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Rohrlich has conjectured that the multiplicative relations in $\mathbb{C}^\times / \overline{\mathbb{Q}}^\times$ between values of $\Gamma$ at rational numbers are generated by the multiplication formula and the reflection formula. In conceptual terms, Lang says that $\Gamma$ is an odd punctured distribution on $\mathbb{Q}/\mathbb{Z}$, and that conjecturally, it is universal, see Relations de distributions et exemples classiques, Deligne's article mentioned in Felipe Voloch's answer and this MO answer. For more details about distributions, see the book by Kubert and Lang, Modular units (Springer, 1981), Chapter 1.

On the other hand, the first Bernoulli polynomial $B_1(x)=x-\frac12$ also gives rise to a universal distribution. More precisely, the function \begin{equation*} h_1 \colon x \mapsto \begin{cases} B_1(\{x\}) & \textrm{if } x \not\in \mathbb{Z} \\ 0 & \textrm{if } x \in \mathbb{Z}, \end{cases} \end{equation*} where $\{x\}=x-\lfloor x \rfloor$ is the fractional part of $x$, is a distribution on $\mathbb{Q}/\mathbb{Z}$. Following the terminology of Kubert-Lang, the first Bernoulli distribution on $\mathbb{Q}/\mathbb{Z}$ is the Stickelberger distribution associated to $h_1$. It takes values in the direct limit of the group rings $\mathbb{Q}[(\mathbb{Z}/N\mathbb{Z})^\times]$. At each finite level $N$, it is given by \begin{align*} \mathbf{B}_1 \colon \bigl(\frac{1}{N}\mathbb{Z}\bigr)/\mathbb{Z} & \to \mathbb{Q}[(\mathbb{Z}/N\mathbb{Z})^\times] \\ x & \mapsto \sum_{u \in (\mathbb{Z}/N\mathbb{Z})^\times} h_1(ux) [u]. \end{align*} The good thing is that we can show that $\mathbf{B}_1$ is universal (among odd punctured distributions on $\mathbb{Q}/\mathbb{Z}$). This follows from the non-vanishing of the Dirichlet $L$-values $L(\chi,1)$ with $\chi$ odd. Concretely, this means that there are no other linear relations than distribution and parity.

This leads to a conjectural criterion for an arbitrary product of $\Gamma$-values \begin{equation*} \Gamma(x_1)^{n_1} \cdots \Gamma(x_r)^{n_r} \end{equation*} with $x_i \in \mathbb{Q} \backslash \mathbb{Z}$ and $n_i \in \mathbb{Z}$, to be an algebraic number times a power of $\sqrt{\pi}$. Namely, just check whether the divisor $X = \sum_{i=1}^r n_i [x_i]$ on $\mathbb{Q}/\mathbb{Z}$ is in the kernel of the first Bernoulli distribution $\mathbf{B}_1$.

If it is, then using linear algebra, you will be able to write $X$ as a linear combination of the multiplication and reflection relations, because $\mathbf{B}_1$ is universal. Therefore you will be able to compute the $\Gamma$ product as an explicit algebraic number times a power of $\sqrt{\pi}$. This algebraic number will be a cyclotomic number times a product of fractional powers of prime numbers.

If one excepts the possible simplifications of this algebraic number, all this can be made into an algorithm.

In the same article, Lang also asks whether the multiplication and reflection formulas generate the ideal of polynomial relations between $\Gamma$-values over $\overline{\mathbb{Q}}(\sqrt{\pi})$. One could try, similarly as above, to give an explicit conjectural criterion for a given polynomial in $\Gamma$-values to be algebraic.

Rohrlich has conjectured that the multiplicative relations in $\mathbb{C}^\times / \overline{\mathbb{Q}}^\times$ between values of $\Gamma$ at rational numbers are generated by the multiplication formula and the reflection formula. In conceptual terms, Lang says that $\Gamma$ is an odd punctured distribution on $\mathbb{Q}/\mathbb{Z}$, and that conjecturally, it is universal, see Relations de distributions et exemples classiques, Deligne's article mentioned in Felipe Voloch's answer and this MO answer. For more details about distributions, see the book by Kubert and Lang, Modular units (Springer, 1981), Chapter 1.

On the other hand, the first Bernoulli polynomial $B_1(x)=x-\frac12$ also gives rise to a universal distribution. More precisely, the function \begin{equation*} h_1 \colon x \mapsto \begin{cases} B_1(\{x\}) & \textrm{if } x \not\in \mathbb{Z} \\ 0 & \textrm{if } x \in \mathbb{Z}, \end{cases} \end{equation*} where $\{x\}=x-\lfloor x \rfloor$ is the fractional part of $x$, is a distribution on $\mathbb{Q}/\mathbb{Z}$. Following the terminology of Kubert-Lang, the first Bernoulli distribution on $\mathbb{Q}/\mathbb{Z}$ is the Stickelberger distribution associated to $h_1$. It takes values in the direct limit of the group rings $\mathbb{Q}[(\mathbb{Z}/N\mathbb{Z})^\times]$. At each finite level $N$, it is given by \begin{align*} \mathbf{B}_1 \colon \bigl(\frac{1}{N}\mathbb{Z}\bigr)/\mathbb{Z} & \to \mathbb{Q}[(\mathbb{Z}/N\mathbb{Z})^\times] \\ x & \mapsto \sum_{u \in (\mathbb{Z}/N\mathbb{Z})^\times} h_1(ux) [u]. \end{align*} The good thing is that we can show that $\mathbf{B}_1$ is universal (among odd punctured distributions on $\mathbb{Q}/\mathbb{Z}$). This follows from the non-vanishing of the Dirichlet $L$-values $L(\chi,1)$ with $\chi$ odd. Concretely, this means that there are no other linear relations than distribution and parity.

This leads to a conjectural criterion for an arbitrary product of $\Gamma$-values \begin{equation*} \Gamma(x_1)^{n_1} \cdots \Gamma(x_r)^{n_r} \end{equation*} with $x_i \in \mathbb{Q} \backslash \mathbb{Z}$ and $n_i \in \mathbb{Z}$, to be an algebraic number times a power of $\sqrt{\pi}$. Namely, just check whether the divisor $X = \sum_{i=1}^r n_i [x_i]$ on $\mathbb{Q}/\mathbb{Z}$ is in the kernel of the first Bernoulli distribution $\mathbf{B}_1$.

If it is, then using linear algebra, you will be able to write $X$ as a linear combination of the multiplication and reflection relations, because $\mathbf{B}_1$ is universal. Therefore you will be able to compute the $\Gamma$ product as an explicit algebraic number times a power of $\sqrt{\pi}$. This algebraic number will be a cyclotomic number times a product of fractional powers of prime numbers.

If one excepts the possible simplifications of this algebraic number, all this can be made into an algorithm.

In the same article, Lang also asks whether the multiplication and reflection formulas generate the ideal of polynomial relations between $\Gamma$-values over $\overline{\mathbb{Q}}(\sqrt{\pi})$. One could try, similarly as above, to give an explicit conjectural criterion for a given polynomial in $\Gamma$-values to be algebraic.

Rohrlich has conjectured that the multiplicative relations in $\mathbb{C}^\times / \overline{\mathbb{Q}}^\times$ between values of $\Gamma$ at rational numbers are generated by the multiplication formula and the reflection formula. In conceptual terms, Lang says that $\Gamma$ is an odd punctured distribution on $\mathbb{Q}/\mathbb{Z}$, and that conjecturally, it is universal, see Relations de distributions et exemples classiques, Deligne's article mentioned in Felipe Voloch's answer and this MO answer. For more details about distributions, see the book by Kubert and Lang, Modular units (Springer, 1981), Chapter 1.

On the other hand, the first Bernoulli polynomial $B_1(x)=x-\frac12$ also gives rise to a universal distribution. More precisely, the function \begin{equation*} h_1 \colon x \mapsto \begin{cases} B_1(\{x\}) & \textrm{if } x \not\in \mathbb{Z} \\ 0 & \textrm{if } x \in \mathbb{Z}, \end{cases} \end{equation*} where $\{x\}=x-\lfloor x \rfloor$ is the fractional part of $x$, is a distribution on $\mathbb{Q}/\mathbb{Z}$. Following the terminology of Kubert-Lang, the first Bernoulli distribution on $\mathbb{Q}/\mathbb{Z}$ is the Stickelberger distribution associated to $h_1$. It takes values in the direct limit of the group rings $\mathbb{Q}[(\mathbb{Z}/N\mathbb{Z})^\times]$. At each finite level $N$, it is given by \begin{align*} \mathbf{B}_1 \colon \bigl(\frac{1}{N}\mathbb{Z}\bigr)/\mathbb{Z} & \to \mathbb{Q}[(\mathbb{Z}/N\mathbb{Z})^\times] \\ x & \mapsto \sum_{u \in (\mathbb{Z}/N\mathbb{Z})^\times} h_1(u) [u]. \end{align*} The good thing is that we can show that $\mathbf{B}_1$ is universal (among odd punctured distributions on $\mathbb{Q}/\mathbb{Z}$). This follows from the non-vanishing of the Dirichlet $L$-values $L(\chi,1)$ with $\chi$ odd. Concretely, this means that there are no other linear relations than distribution and parity.

This leads to a conjectural criterion for an arbitrary product of $\Gamma$-values \begin{equation*} \Gamma(x_1)^{n_1} \cdots \Gamma(x_r)^{n_r} \end{equation*} with $x_i \in \mathbb{Q} \backslash \mathbb{Z}$ and $n_i \in \mathbb{Z}$, to be an algebraic number times a power of $\sqrt{\pi}$. Namely, just check whether the divisor $X = \sum_{i=1}^r n_i [x_i]$ on $\mathbb{Q}/\mathbb{Z}$ is in the kernel of the first Bernoulli distribution $\mathbf{B}_1$.

If it is, then using linear algebra, you will be able to write $X$ as a linear combination of the multiplication and reflection relations, because $\mathbf{B}_1$ is universal. Therefore you will be able to compute the $\Gamma$ product as an explicit algebraic number times a power of $\sqrt{\pi}$. This algebraic number will be a cyclotomic number times a product of fractional powers of prime numbers.

If one excepts the possible simplifications of this algebraic number, all this can be made into an algorithm.

In the same article, Lang also asks whether the multiplication and reflection formulas generate the ideal of polynomial relations between $\Gamma$-values over $\overline{\mathbb{Q}}(\sqrt{\pi})$. One could try, similarly as above, to give an explicit conjectural criterion for a given polynomial in $\Gamma$-values to be algebraic.

Rohrlich has conjectured that the multiplicative relations in $\mathbb{C}^\times / \overline{\mathbb{Q}}^\times$ between values of $\Gamma$ at rational numbers are generated by the multiplication formula and the reflection formula. In conceptual terms, Lang says that $\Gamma$ is an odd punctured distribution on $\mathbb{Q}/\mathbb{Z}$, and that conjecturally, it is universal, see Relations de distributions et exemples classiques, Deligne's article mentioned in Felipe Voloch's answer and this MO answer. For more details about distributions, see the book by Kubert and Lang, Modular units (Springer, 1981), Chapter 1.

On the other hand, the first Bernoulli polynomial $B_1(x)=x-\frac12$ also gives rise to a universal distribution. More precisely, the function \begin{equation*} h_1 \colon x \mapsto \begin{cases} B_1(\{x\}) & \textrm{if } x \not\in \mathbb{Z} \\ 0 & \textrm{if } x \in \mathbb{Z}, \end{cases} \end{equation*} where $\{x\}=x-\lfloor x \rfloor$ is the fractional part of $x$, is a distribution on $\mathbb{Q}/\mathbb{Z}$. Following the terminology of Kubert-Lang, the first Bernoulli distribution on $\mathbb{Q}/\mathbb{Z}$ is the Stickelberger distribution associated to $h_1$. It takes values in the direct limit of the group rings $\mathbb{Q}[(\mathbb{Z}/N\mathbb{Z})^\times]$. At each finite level $N$, it is given by \begin{align*} \mathbf{B}_1 \colon \bigl(\frac{1}{N}\mathbb{Z}\bigr)/\mathbb{Z} & \to \mathbb{Q}[(\mathbb{Z}/N\mathbb{Z})^\times] \\ x & \mapsto \sum_{u \in (\mathbb{Z}/N\mathbb{Z})^\times} h_1(ux) [u]. \end{align*} The good thing is that we can show that $\mathbf{B}_1$ is universal (among odd punctured distributions on $\mathbb{Q}/\mathbb{Z}$). This follows from the non-vanishing of the Dirichlet $L$-values $L(\chi,1)$ with $\chi$ odd. Concretely, this means that there are no other linear relations than distribution and parity.

This leads to a conjectural criterion for an arbitrary product of $\Gamma$-values \begin{equation*} \Gamma(x_1)^{n_1} \cdots \Gamma(x_r)^{n_r} \end{equation*} with $x_i \in \mathbb{Q} \backslash \mathbb{Z}$ and $n_i \in \mathbb{Z}$, to be an algebraic number times a power of $\sqrt{\pi}$. Namely, just check whether the divisor $X = \sum_{i=1}^r n_i [x_i]$ on $\mathbb{Q}/\mathbb{Z}$ is in the kernel of the first Bernoulli distribution $\mathbf{B}_1$.

If it is, then using linear algebra, you will be able to write $X$ as a linear combination of the multiplication and reflection relations, because $\mathbf{B}_1$ is universal. Therefore you will be able to compute the $\Gamma$ product as an explicit algebraic number times a power of $\sqrt{\pi}$. This algebraic number will be a cyclotomic number times a product of fractional powers of prime numbers.

If one excepts the possible simplifications of this algebraic number, all this can be made into an algorithm.

In the same article, Lang also asks whether the multiplication and reflection formulas generate the ideal of polynomial relations between $\Gamma$-values over $\overline{\mathbb{Q}}(\sqrt{\pi})$. One could try, similarly as above, to give an explicit conjectural criterion for a given polynomial in $\Gamma$-values to be algebraic.

Rohrlich has conjectured that the multiplicative relations in $\mathbb{C}^\times / \overline{\mathbb{Q}}^\times$ between values of $\Gamma$ at rational numbers are generated by the multiplication formula and the reflection formula. In conceptual terms, Lang says that $\Gamma$ is an odd punctured distribution on $\mathbb{Q}/\mathbb{Z}$, and that conjecturally, it is universal, see Relations de distributions et exemples classiques, Deligne's article mentioned in Felipe Voloch's answer and this MO answer.

On the other hand, the first Bernoulli polynomial $B_1(x)$ also provides an odd punctured distribution on $\mathbb{Q}/\mathbb{Z}$, and we know it is universal (this follows from the non-vanishing of the Dirichlet $L$-values $L(\chi,1)$ with $\chi$ odd). This leads to a conjectural criterion for an arbitrary product of $\Gamma$-values \begin{equation*} \Gamma(x_1)^{n_1} \cdots \Gamma(x_r)^{n_r} \end{equation*} with $x_i \in \mathbb{Q} \backslash \mathbb{Z}$ and $n_i \in \mathbb{Z}$, to be an algebraic number times a power of $\sqrt{\pi}$. Namely, just check whether the divisor $X = \sum_{i=1}^r n_i [x_i]$ on $\mathbb{Q}/\mathbb{Z}$ is in the kernel of the Bernoulli distribution.

If it is, then using linear algebra, you will be able to write $X$ as a linear combination of the multiplication and reflection relations, because the Bernoulli distribution is universal. Therefore you will be able to compute the $\Gamma$ product as an explicit algebraic number times a power of $\sqrt{\pi}$. This algebraic number will be a cyclotomic number times a product of fractional powers of prime numbers.

If one excepts the possible simplifications of this algebraic number, all this can be made into an algorithm.

In the same article, Lang also asks whether the multiplication and reflection formulas generate the ideal of polynomial relations between $\Gamma$-values over $\overline{\mathbb{Q}}(\sqrt{\pi})$. One could try, similarly as above, to give an explicit conjectural criterion for a given polynomial in $\Gamma$-values to be algebraic.

Rohrlich has conjectured that the multiplicative relations in $\mathbb{C}^\times / \overline{\mathbb{Q}}^\times$ between values of $\Gamma$ at rational numbers are generated by the multiplication formula and the reflection formula. In conceptual terms, Lang says that $\Gamma$ is an odd punctured distribution on $\mathbb{Q}/\mathbb{Z}$, and that conjecturally, it is universal, see Relations de distributions et exemples classiques, Deligne's article mentioned in Felipe Voloch's answer and this MO answer. For more details about distributions, see the book by Kubert and Lang, Modular units (Springer, 1981), Chapter 1.

On the other hand, the first Bernoulli polynomial $B_1(x)=x-\frac12$ also gives rise to a universal distribution. More precisely, the function \begin{equation*} h_1 \colon x \mapsto \begin{cases} B_1(\{x\}) & \textrm{if } x \not\in \mathbb{Z} \\ 0 & \textrm{if } x \in \mathbb{Z}, \end{cases} \end{equation*} where $\{x\}=x-\lfloor x \rfloor$ is the fractional part of $x$, is a distribution on $\mathbb{Q}/\mathbb{Z}$. Following the terminology of Kubert-Lang, the first Bernoulli distribution on $\mathbb{Q}/\mathbb{Z}$ is the Stickelberger distribution associated to $h_1$. It takes values in the direct limit of the group rings $\mathbb{Q}[(\mathbb{Z}/N\mathbb{Z})^\times]$. At each finite level $N$, it is given by \begin{align*} \mathbf{B}_1 \colon \bigl(\frac{1}{N}\mathbb{Z}\bigr)/\mathbb{Z} & \to \mathbb{Q}[(\mathbb{Z}/N\mathbb{Z})^\times] \\ x & \mapsto \sum_{u \in (\mathbb{Z}/N\mathbb{Z})^\times} h_1(u) [u]. \end{align*} The good thing is that we can show that $\mathbf{B}_1$ is universal (among odd punctured distributions on $\mathbb{Q}/\mathbb{Z}$). This follows from the non-vanishing of the Dirichlet $L$-values $L(\chi,1)$ with $\chi$ odd. Concretely, this means that there are no other linear relations than distribution and parity. 

This leads to a conjectural criterion for an arbitrary product of $\Gamma$-values \begin{equation*} \Gamma(x_1)^{n_1} \cdots \Gamma(x_r)^{n_r} \end{equation*} with $x_i \in \mathbb{Q} \backslash \mathbb{Z}$ and $n_i \in \mathbb{Z}$, to be an algebraic number times a power of $\sqrt{\pi}$. Namely, just check whether the divisor $X = \sum_{i=1}^r n_i [x_i]$ on $\mathbb{Q}/\mathbb{Z}$ is in the kernel of the first Bernoulli distribution $\mathbf{B}_1$.

If it is, then using linear algebra, you will be able to write $X$ as a linear combination of the multiplication and reflection relations, because $\mathbf{B}_1$ is universal. Therefore you will be able to compute the $\Gamma$ product as an explicit algebraic number times a power of $\sqrt{\pi}$. This algebraic number will be a cyclotomic number times a product of fractional powers of prime numbers.

If one excepts the possible simplifications of this algebraic number, all this can be made into an algorithm.

In the same article, Lang also asks whether the multiplication and reflection formulas generate the ideal of polynomial relations between $\Gamma$-values over $\overline{\mathbb{Q}}(\sqrt{\pi})$. One could try, similarly as above, to give an explicit conjectural criterion for a given polynomial in $\Gamma$-values to be algebraic.