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

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- by Shigeki Akiyama and Yoshio Tanigawa PDF
- Math. Comp.
**68**(1999), 1201-1231 Request permission

## 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.## References

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## Additional Information

**Shigeki Akiyama**- Affiliation: Department of Mathematics, Faculty of Science, Niigata University, Ikarashi 2-8050, Niigata 950-2181, Japan
- Email: akiyama@math.sc.niigata-u.ac.jp
**Yoshio Tanigawa**- Affiliation: Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan
- Email: tanigawa@math.nagoya-u.ac.jp
- Received by editor(s): May 22, 1996
- Received by editor(s) in revised form: December 11, 1996
- Published electronically: February 10, 1999
- © Copyright 1999 American Mathematical Society
- Journal: Math. Comp.
**68**(1999), 1201-1231 - MSC (1991): Primary 11F11, 11G40, 11M26
- DOI: https://doi.org/10.1090/S0025-5718-99-01051-0
- MathSciNet review: 1627842