A note on $L(1)$ of Hecke $L-$series associated to the elliptic curves with CM by $\sqrt{-3}$

Abstract

Consider elliptic curves $E:y^2=x^3+D^3$ defined over the quadratic field $\ \Q(\sqrt{-3}) $. Hecke $L-$series attached to $E$ are studied, formulae for their values at $s=1,$ and bound of 3-adic valuations of these values are given. These results are complementary to those in [Q] and [QZ], and are consistent with the predictions of the conjecture of Birch and Swinnerton-Dyer.

arXiv comment11 pages