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

TL;DR

This paper investigates the properties of Hecke L-series linked to elliptic curves with complex multiplication by b1a7, providing explicit formulas and valuations at s=1, supporting the Birch and Swinnerton-Dyer conjecture.

Contribution

It offers new explicit formulas for L-series values at s=1 and bounds on their 3-adic valuations for elliptic curves with CM by b1a7.

Findings

Formulas for L-series values at s=1

Bounds on 3-adic valuations of these values

Results align with BSD conjecture predictions

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.