{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:41:28Z","timestamp":1776724888824,"version":"3.51.2"},"reference-count":25,"publisher":"American Mathematical Society (AMS)","issue":"227","license":[{"start":{"date-parts":[[2000,2,10]],"date-time":"2000-02-10T00:00:00Z","timestamp":950140800000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We calculated numerically the values of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L\">\n                        <mml:semantics>\n                          <mml:mi>L<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">L<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -functions of four typical elliptic curves in the critical strip in the range\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"Im left-parenthesis s right-parenthesis less-than-or-equal-to 400\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mtext>Im<\/mml:mtext>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>s<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mn>400<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\text {Im}(s)\\leq 400<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . We found that all the non-trivial zeros in this range lie on the critical line\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"Re left-parenthesis s right-parenthesis equals 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mtext>Re<\/mml:mtext>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>s<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\text {Re}(s)=1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and are simple except the one at\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"s equals 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>s<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">s=1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . 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\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"Re left-parenthesis s right-parenthesis greater-than 0\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mtext>Re<\/mml:mtext>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>s<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\text {Re}(s)&gt;0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , which enables us to calculate for large\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"Im left-parenthesis s right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mtext>Im<\/mml:mtext>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>s<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\text {Im}(s)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . Furthermore we remark that a relation exists between Sato-Tate conjecture and the generalized Riemann Hypothesis.\n                  <\/p>","DOI":"10.1090\/s0025-5718-99-01051-0","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:13:53Z","timestamp":1027707233000},"page":"1201-1231","source":"Crossref","is-referenced-by-count":19,"title":["Calculation of values of \ud835\udc3f-functions associated to elliptic curves"],"prefix":"10.1090","volume":"68","author":[{"given":"Shigeki","family":"Akiyama","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yoshio","family":"Tanigawa","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[1999,2,10]]},"reference":[{"key":"1","isbn-type":"print","volume-title":"Algorithms for modular elliptic curves","author":"Cremona, J. E.","year":"1992","ISBN":"https:\/\/id.crossref.org\/isbn\/0521418135"},{"key":"2","unstructured":"F. Diamond, On deformation rings and Hecke rings, preprint."},{"key":"3","series-title":"Pure and Applied Mathematics, Vol. 58","volume-title":"Riemann's zeta function","author":"Edwards, H. M.","year":"1974"},{"issue":"2","key":"4","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1080\/10586458.1992.10504254","article-title":"Z\u00e9ros des fonctions \ud835\udc3f de courbes elliptiques","volume":"1","author":"Fermigier, St\u00e9fane","year":"1992","journal-title":"Experiment. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1058-6458","issn-type":"print"},{"key":"5","series-title":"Encyclopedia of Mathematics and its Applications","isbn-type":"print","volume-title":"Continued fractions","volume":"11","author":"Jones, William B.","year":"1980","ISBN":"https:\/\/id.crossref.org\/isbn\/0201135108"},{"key":"6","doi-asserted-by":"publisher","first-page":"693","DOI":"10.1137\/0708063","article-title":"A posteriori bounds for the truncation error of continued fractions","volume":"8","author":"Jones, William B.","year":"1971","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"7","unstructured":"S. Hitotumatu, J. Yamauchi and T. Uno,S\u00fbchikeisanhou III (Numerical Computing Methods III), Baihukan, 1971 (Japanese)."},{"key":"8","unstructured":"T. Kano (ed.) Riemann yosou (Riemann Hypothesis), Nihonhyouronsha, 1991 (Japanese)."},{"key":"9","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-58018-5","volume-title":"Basic analytic number theory","author":"Karatsuba, Anatolij A.","year":"1993","ISBN":"https:\/\/id.crossref.org\/isbn\/3540533451"},{"key":"10","series-title":"Mathematical Notes","isbn-type":"print","volume-title":"Elliptic curves","volume":"40","author":"Knapp, Anthony W.","year":"1992","ISBN":"https:\/\/id.crossref.org\/isbn\/0691085595"},{"key":"11","series-title":"Pure and Applied Mathematics","volume-title":"Uniform distribution of sequences","author":"Kuipers, L.","year":"1974"},{"key":"12","first-page":"134","article-title":"Approximate functional equations of Dirichlet functions","volume":"32","author":"Lavrik, A. F.","year":"1968","journal-title":"Izv. Akad. Nauk SSSR Ser. Mat.","ISSN":"https:\/\/id.crossref.org\/issn\/0373-2436","issn-type":"print"},{"issue":"174","key":"13","doi-asserted-by":"publisher","first-page":"667","DOI":"10.2307\/2008005","article-title":"On the zeros of the Riemann zeta function in the critical strip. IV","volume":"46","author":"van de Lune, J.","year":"1986","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"6","key":"14","first-page":"7","article-title":"Cyclotomic fields and modular curves","volume":"26","author":"Manin, Ju. I.","year":"1971","journal-title":"Uspehi Mat. Nauk","ISSN":"https:\/\/id.crossref.org\/issn\/0042-1316","issn-type":"print"},{"key":"15","first-page":"19","article-title":"Parabolic points and zeta functions of modular curves","volume":"36","author":"Manin, Ju. I.","year":"1972","journal-title":"Izv. Akad. Nauk SSSR Ser. Mat.","ISSN":"https:\/\/id.crossref.org\/issn\/0373-2436","issn-type":"print"},{"key":"16","unstructured":"A.M. Odlyzko, The 10\u00b2\u2070-th Zeros of the Riemann Zeta Function and 70 Million of its Neighbors, preprint"},{"key":"17","doi-asserted-by":"publisher","first-page":"198","DOI":"10.1007\/BF01404324","article-title":"A remark on the Sato-Tate conjecture","volume":"9","author":"Ogg, A. P.","year":"1969","journal-title":"Invent. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0020-9910","issn-type":"print"},{"key":"18","isbn-type":"print","doi-asserted-by":"publisher","first-page":"159","DOI":"10.1090\/crmp\/004\/11","article-title":"Symmetric power \ud835\udc3f-functions for \ud835\udc3a\ud835\udc3f(2)","author":"Shahidi, Freydoon","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/0821869949"},{"issue":"3","key":"19","doi-asserted-by":"publisher","first-page":"553","DOI":"10.2307\/2118560","article-title":"Ring-theoretic properties of certain Hecke algebras","volume":"141","author":"Taylor, Richard","year":"1995","journal-title":"Ann. of Math. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-486X","issn-type":"print"},{"key":"20","isbn-type":"print","volume-title":"The theory of the Riemann zeta-function","author":"Titchmarsh, E. C.","year":"1986","ISBN":"https:\/\/id.crossref.org\/isbn\/0198533691","edition":"2"},{"issue":"5","key":"21","first-page":"49","article-title":"An approximate functional equation and moments of the Dirichlet series generated by the Ramanujan function","author":"Turganaliev, R. T.","year":"1992","journal-title":"Izv. Akad. Nauk Respub. Kazakhstan Ser. Fiz.-Mat.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-3191","issn-type":"print"},{"key":"22","doi-asserted-by":"publisher","first-page":"783","DOI":"10.2307\/2371336","article-title":"Ring homomorphisms which are also lattice homomorphisms","volume":"61","author":"Ward, Morgan","year":"1939","journal-title":"Amer. J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9327","issn-type":"print"},{"issue":"3","key":"23","doi-asserted-by":"publisher","first-page":"443","DOI":"10.2307\/2118559","article-title":"Modular elliptic curves and Fermat\u2019s last theorem","volume":"141","author":"Wiles, Andrew","year":"1995","journal-title":"Ann. of Math. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-486X","issn-type":"print"},{"issue":"1","key":"24","first-page":"87","article-title":"On calculations of zeros of \ud835\udc3f-functions related with Ramanujan\u2019s discriminant function on the critical line","volume":"3","author":"Yoshida, Hiroyuki","year":"1988","journal-title":"J. Ramanujan Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0970-1249","issn-type":"print"},{"key":"25","unstructured":"\\bysame, On calculations of zeros of various L-functions, Symposium on automorphic forms at Kinosaki (1993), 47-72."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/1999-68-227\/S0025-5718-99-01051-0\/S0025-5718-99-01051-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/1999-68-227\/S0025-5718-99-01051-0\/S0025-5718-99-01051-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:06:02Z","timestamp":1776722762000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/1999-68-227\/S0025-5718-99-01051-0\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1999,2,10]]},"references-count":25,"journal-issue":{"issue":"227","published-print":{"date-parts":[[1999,7]]}},"alternative-id":["S0025-5718-99-01051-0"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-99-01051-0","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[1999,2,10]]}}}