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Let $F(s) =\sum_{n=1}^\infty a_n n^{-s}= \sum_{j=1}^J c_j F_j(s)$ be a linear combination of L-functions $F_j$ of degree $\le d$, that is $(s-1)^r F_j(s)$ entire and its functional equation contains at most $d$ gamma factors $\Gamma(s/2+a)$.

Then for $m > d$ or for $m=d,|x| < C$, $\Gamma(ms)F(s)x^{-s}$ decays fast enough to apply the residue theorem $$\begin{eqnarray}f_m(x)&=&\frac{1}{2i\pi} \int_{2-i\infty}^{2+i\infty} F(s) \Gamma(ms) x^{-s}ds \\ &=& \sum Res(F(s) \Gamma(ms) x^{-s}) \\ &=& Res(F(s),1) \Gamma(m) x^{-1}+ \sum_{k=0}^\infty \frac{(-1)^k}{k!m} F(-k/m) x^{k/m}\end{eqnarray}$$ (for $m = d$ it is valid only for $|x| < C$ but the exponential decay of $\Gamma(ms)F(s)$ on $\Im(s)=2$ implies $f_m$ is analytic so it is determined by $x \in (0,C)$)

And hence $$F(s) = \frac{1}{\Gamma(ms)} \int_0^\infty f_m(x) x^{s-1}dx$$

is fully determined by its residue at $1$ and its values at $s=-k/m$.

In other words your claim holds only for L-functions of degree $d=1$.

Let $F(s) =\sum_{n=1}^\infty a_n n^{-s}= \sum_{j=1}^J c_j F_j(s)$ be a linear combination of L-functions $F_j$ of degree $\le d$, that is $(s-1)^r F_j(s)$ entire and its functional equation contains at most $d$ gamma factors $\Gamma(s/2+a)$.

Then for $m > d$ or for $m=d,|x| < C$, $\Gamma(ms)F(s)x^{-s}$ decays fast enough to apply the residue theorem $$\begin{eqnarray}f_m(x)&=&\frac{1}{2i\pi} \int_{2-i\infty}^{2+i\infty} F(s) \Gamma(ms) x^{-s}ds \\ &=& \sum Res(F(s) \Gamma(ms) x^{-s}) \\ &=& \sum Res_{s=1}(F(s) \Gamma(ms)x^{-s}) + \sum_{k=0}^\infty \frac{(-1)^k}{k!m} F(-k/m) x^{k/m}\end{eqnarray}$$ (for $m = d$ it is valid only for $|x| < C$ but the exponential decay of $\Gamma(ms)F(s)$ on $\Im(s)=2$ implies $f_m$ is analytic so it is determined by $x \in (0,C)$)

And hence $$F(s) = \frac{1}{\Gamma(ms)} \int_0^\infty f_m(x) x^{s-1}dx$$

is fully determined by its principal part at $1$ and its values at $s=-k/m$.

In other words your claim holds only for L-functions of degree $d=1$.

Let $F(s) =\sum_{n=1}^\infty a_n n^{-s}= \sum_{j=1}^J c_j F_j(s)$ be a linear combination of L-functions $F_j$ of degree $\le d$, that is $(s-1)^r F_j(s)$ entire and its functional equation contains at most $d$ gamma factors $\Gamma(s/2)$ or $\Gamma((s+1)/2)$.

Then for $m \ge d$, $\Gamma(ms)F(s)x^{-s}$ decays fast enough to apply the residue theorem $$\begin{eqnarray}f_m(x)&=&\frac{1}{2i\pi} \int_{2-i\infty}^{2+i\infty} F(s) \Gamma(ms) x^{-s}ds \\ &=& \sum Res(F(s) \Gamma(ms) x^{-s}) \\ &=& Res(F(s),1) \Gamma(m) x^{-1}+ \sum_{k=0}^\infty \frac{(-1)^k}{k!m} F(-k/m) x^{k/m}\end{eqnarray}$$ (for $m > d$ it is valid for every $x$, for $m = d$ it is valid only for $x$ small)

And hence $$F(s) = \frac{1}{\Gamma(ms)} \int_0^\infty f_m(x) x^{s-1}dx$$

is fully determined by its residue at $1$ and its values at $s=-k/m$.

In other words your claim holds only for L-functions of degree $d=1$.

Let $F(s) =\sum_{n=1}^\infty a_n n^{-s}= \sum_{j=1}^J c_j F_j(s)$ be a linear combination of L-functions $F_j$ of degree $\le d$, that is $(s-1)^r F_j(s)$ entire and its functional equation contains at most $d$ gamma factors $\Gamma(s/2+a)$.

Then for $m > d$ or for $m=d,|x| < C$, $\Gamma(ms)F(s)x^{-s}$ decays fast enough to apply the residue theorem $$\begin{eqnarray}f_m(x)&=&\frac{1}{2i\pi} \int_{2-i\infty}^{2+i\infty} F(s) \Gamma(ms) x^{-s}ds \\ &=& \sum Res(F(s) \Gamma(ms) x^{-s}) \\ &=& Res(F(s),1) \Gamma(m) x^{-1}+ \sum_{k=0}^\infty \frac{(-1)^k}{k!m} F(-k/m) x^{k/m}\end{eqnarray}$$ (for $m = d$ it is valid only for $|x| < C$ but the exponential decay of $\Gamma(ms)F(s)$ on $\Im(s)=2$ implies $f_m$ is analytic so it is determined by $x \in (0,C)$)

And hence $$F(s) = \frac{1}{\Gamma(ms)} \int_0^\infty f_m(x) x^{s-1}dx$$

is fully determined by its residue at $1$ and its values at $s=-k/m$.

In other words your claim holds only for L-functions of degree $d=1$.

Let $F(s) =\sum_{n=1}^\infty a_n n^{-s}= \sum_{j=1}^J c_j F_j(s)$ be a linear combination of L-functions $F_j$ of degree $\le d$, that is $(s-1)^r F_j(s)$ entire and its functional equation contains at most $d$ gamma factors $\Gamma(s/2)$ or $\Gamma((s+1)/2)$.

Then for $m \ge d$, $\Gamma(ms)F(s)$ decays fast enough to apply the residue theorem to its inverse Mellin transform $$\begin{eqnarray}f_m(x)&=&\frac{1}{2i\pi} \int_{2-i\infty}^{2+i\infty} F(s) \Gamma(ms) x^{-s}ds \\ &=& \sum Res(F(s) \Gamma(ms) x^{-s}) \\ &=& Res(F(s),1) \Gamma(m) x^{-1}+ \sum_{k=0}^\infty \frac{(-1)^k}{k!m} F(-k/m) x^{k/m}\end{eqnarray}$$ (for $m > d$ it is valid for every $x$, for $m = d$ it is valid only for $x$ small)

And hence $$F(s) = \frac{1}{\Gamma(ms)} \int_0^\infty f_m(x) x^{s-1}dx$$

is fully determined by its residue at $1$ and its values at $s=-k/m$.

In other words your claim holds only for L-functions of degree $d=1$.

Let $F(s) =\sum_{n=1}^\infty a_n n^{-s}= \sum_{j=1}^J c_j F_j(s)$ be a linear combination of L-functions $F_j$ of degree $\le d$, that is $(s-1)^r F_j(s)$ entire and its functional equation contains at most $d$ gamma factors $\Gamma(s/2)$ or $\Gamma((s+1)/2)$.

Then for $m \ge d$, $\Gamma(ms)F(s)x^{-s}$ decays fast enough to apply the residue theorem $$\begin{eqnarray}f_m(x)&=&\frac{1}{2i\pi} \int_{2-i\infty}^{2+i\infty} F(s) \Gamma(ms) x^{-s}ds \\ &=& \sum Res(F(s) \Gamma(ms) x^{-s}) \\ &=& Res(F(s),1) \Gamma(m) x^{-1}+ \sum_{k=0}^\infty \frac{(-1)^k}{k!m} F(-k/m) x^{k/m}\end{eqnarray}$$ (for $m > d$ it is valid for every $x$, for $m = d$ it is valid only for $x$ small)

And hence $$F(s) = \frac{1}{\Gamma(ms)} \int_0^\infty f_m(x) x^{s-1}dx$$

is fully determined by its residue at $1$ and its values at $s=-k/m$.

In other words your claim holds only for L-functions of degree $d=1$.