Formula and 2-adic valuation of L(1) of elliptic curves with CM by -3

摘要

For the rational integers λ≡1, 3, or 5 (mod 6), considering elliptic curves y2＝x3-2433Dλ over the field (-3), the formula for the value at s=1 of Hecke L-series attached to such elliptic curves, expressed as a finite sum of values of Weierstrass -functions, is obtained. Moreover, when λ≡3 (mod 6), the lower bounds of 2-adic valuations of these values are also obtained. These results are consistent with the predictions of the conjecture of Birch and Swinnerton-Dyer in a sense, and have generalized and advanced some results in recent literature.