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On the conjecture of Birch and Swinnerton-Dyer for an elliptic curve of rank $3$

Authors: Joe P. Buhler, Benedict H. Gross and Don B. Zagier
Journal: Math. Comp. 44 (1985), 473-481
MSC: Primary 11G40; Secondary 14G25
MathSciNet review: 777279
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Abstract: The elliptic curve ${y^2} = 4{x^3} - 28x + 25$ has rank 3 over Q. Assuming the Weil-Taniyama conjecture for this curve, we show that its L-series $L(s)$ has a triple zero at $s = 1$ and compute ${\lim _{s \to 1}}L(s)/{(s - 1)^3}$ to 28 decimal places; its value agrees with the product of the regulator and real period, in accordance with the Birch-Swinnerton-Dyer conjecture if III is trivial.

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Article copyright: © Copyright 1985 American Mathematical Society