Let

• $E:y^2＝x^3-x$ be an elliptic curve over $ \Bbb{Q}$ and
• $ω_E＝dx/2y＝dx/2\sqrt{x^3-x}$.

Then $$ \begin{split} \Omega(E)&＝\int_{E(\Bbb{R})} ω_E\\ \\ &＝2\int\limits_{1}^{+\infty} dx/\sqrt{x^3-x} \end{split} $$ Question. Let the Hasse-Weil $L$-function be $L(E,1)$.
How can I prove that $$ L(E,1)/\Omega(E)＝1/8\; ? \label{1}\tag{$\star$} $$ If you know any reference (★) is proved, I'll appreciate if you could show me how to calculate (★).

Let

• $E:y^2=x^3-x$ be an elliptic curve over $ \Bbb{Q}$ and
• $ω_E=dx/2y=dx/2\sqrt{x^3-x}$.

Then $$ \begin{split} \Omega(E)&=\int_{E(\Bbb{R})} ω_E\\ \\ &=2\int\limits_1^{+\infty} dx/\sqrt{x^3-x} \end{split} $$ Question. Let the Hasse-Weil $L$-function be $L(E,1)$.
How can I prove that $$ L(E,1)/\Omega(E)=1/8\text{ ?} \label{1}\tag{$\star$} $$ If you know any reference (★) is proved, I'll appreciate if you could show me how to calculate (★).

Let

• $E:y^2＝x^3-x$ be an elliptic curve over $ \Bbb{Q}$ and
• $ω_E＝dx/2y＝dx/2\sqrt{x^3-x}$.

Then $$ \begin{split} \Omega(E)&＝\int_{E(\Bbb{R})} ω_E\\ \\ &＝2\int\limits_{1}^{+\infty} dx/\sqrt{x^3-x} \end{split} $$ Question. Let the Hasse-Weil $L$-function be $L(E,1)$.
How can I prove that $$ L(E,1)/\Omega(E)＝1/8\; ? \label{1}\tag{$\star$} $$ If you know any reference (★) is proved, I'll appreciate if you could share it me. However, a self contained argument would be also appreciated.

Let

• $E:y^2＝x^3-x$ be an elliptic curve over $ \Bbb{Q}$ and
• $ω_E＝dx/2y＝dx/2\sqrt{x^3-x}$.

Then $$ \begin{split} \Omega(E)&＝\int_{E(\Bbb{R})} ω_E\\ \\ &＝2\int\limits_{1}^{+\infty} dx/\sqrt{x^3-x} \end{split} $$ Question. Let the Hasse-Weil $L$-function be $L(E,1)$.
How can I prove that $$ L(E,1)/\Omega(E)＝1/8\; ? \label{1}\tag{$\star$} $$ If you know any reference (★) is proved, I'll appreciate if you could show me how to calculate (★).

Let

• $E:y^2＝x^3-x$ be an elliptic curve over $ \Bbb{Q}$ and
• $ω_E＝dx/2y＝dx/2\sqrt{x^3-x}$.

Then $$ \begin{split} \Omega(E)&＝\int_{E(\Bbb{R})} ω_E\\ \\ &＝2\int\limits_{1}^{+\infty} dx/\sqrt{x^3-x} \end{split} $$ Question. Let the Hasse-Weil $L$-function be $L(E,1)$.
How can I prove that $$ L(E,1)/\Omega(E)＝1/8\; ? \label{1}\tag{$\star$} $$ If you know any reference where \eqref{1} is proved, I'll appreciate if you could share it me. However, a self contained argument would be also appreciated.

Let

• $E:y^2＝x^3-x$ be an elliptic curve over $ \Bbb{Q}$ and
• $ω_E＝dx/2y＝dx/2\sqrt{x^3-x}$.

Then $$ \begin{split} \Omega(E)&＝\int_{E(\Bbb{R})} ω_E\\ \\ &＝2\int\limits_{1}^{+\infty} dx/\sqrt{x^3-x} \end{split} $$ Question. Let the Hasse-Weil $L$-function be $L(E,1)$.
How can I prove that $$ L(E,1)/\Omega(E)＝1/8\; ? \label{1}\tag{$\star$} $$ If you know any reference (★) is proved, I'll appreciate if you could share it me. However, a self contained argument would be also appreciated.