Special values of L-series attached to two families of CM elliptic curves

Consider two families of elliptic curves y2 x3 D1 x and y2＝ x3 4 3 2＝x3-D1x and y2=x3-2433D22 over fields ( --1 )and ( -3 ) respectively. Formulae expressed by Weierstrass _qfunctions are given for special values of Hecke L-se ries attached to such elliptic curves. The lower bounds of 2-adic and 3-adic valuations of these values of Hecke L-series as well as criteria for reaching these bounds are obtained. These results are consistent with the predictions of the conjec ture of Birch and Swinnerton-Dyer; and some results in recent literature for more special case and for 2-adic valuation are also developed.