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

Abstract

The elliptic curve y 2 = 4 x 3 − 28 x + 25 {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 ) L(s) has a triple zero at s = 1 s = 1 and compute lim s → 1 L ( s ) / ( s − 1 ) 3 {\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.