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Mathematics of Computation

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Calculation of values of $L$-functions
associated to elliptic curves

Authors: Shigeki Akiyama and Yoshio Tanigawa
Journal: Math. Comp. 68 (1999), 1201-1231
MSC (1991): Primary 11F11, 11G40, 11M26
Published electronically: February 10, 1999
MathSciNet review: 1627842
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Abstract: We calculated numerically the values of $L$-functions of four typical elliptic curves in the critical strip in the range $\text{Im}(s)\leq 400$. We found that all the non-trivial zeros in this range lie on the critical line $\text{Re}(s)=1$ and are simple except the one at $s=1$. The method we employed in this paper is the approximate functional equation with incomplete gamma functions in the coefficients. For incomplete gamma functions, we continued them holomorphically to the right half plane $\text{Re}(s)>0$, which enables us to calculate for large $\text{Im}(s)$. Furthermore we remark that a relation exists between Sato-Tate conjecture and the generalized Riemann Hypothesis.

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Additional Information

Shigeki Akiyama
Affiliation: Department of Mathematics, Faculty of Science, Niigata University, Ikarashi 2-8050, Niigata 950-2181, Japan

Yoshio Tanigawa
Affiliation: Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan

Keywords: Elliptic curve, $L$-function, approximate functional equation, Sato-Tate conjecture, Riemann Hypothesis
Received by editor(s): May 22, 1996
Received by editor(s) in revised form: December 11, 1996
Published electronically: February 10, 1999
Article copyright: © Copyright 1999 American Mathematical Society