Mathematics of Computation

Author(s): Shigeki Akiyama; Yoshio Tanigawa.
Journal: Math. Comp. 68 (1999), 1201-1231.
MSC (1991): Primary 11F11, 11G40, 11M26
Posted: February 10, 1999
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Abstract: We calculated numerically the values of

-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

. 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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Retrieve articles in Mathematics of Computation with MSC (1991): 11F11, 11G40, 11M26

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

DOI: 10.1090/S0025-5718-99-01051-0
PII: S 0025-5718(99)01051-0
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
Posted: February 10, 1999
Copyright of article: Copyright 1999, American Mathematical Society

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