The first author(2022)proposed a new method to establish the Shimura lift from a cusp form of weight 3/2 to that of weight 2.This method has been successfully applied to congruent number elliptic curves,yielding new results in the congruent number problem.In this paper,we further apply this method to the elliptic curves En:y2=x3+n2x,providing a complete proof of the formula for the values of the L-function of En at 1 and analyzing the conditions for the nonvanishing of L(En,1).The proof is based on some equalities regarding the number of representations by ternary and quaternary quadratic forms.
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