A Criterion for Elliptic Curves with Second Lowest 2-Power in L(1) (II)

Let D = p 1 p 2 · · ·p m , where p 1, p 2, . . . , p m are distinct rational primes with p 1 . p 2 .3(mod 8), p i .1(mod 8)(3 . i . m), and m is any positive integer. In this paper, we give a simple combinatorial criterion for the value of the complex L.function of the congruent elliptic curve ED2:y2=x3.D2xat s = 1, divided by the period . defined below, to be exactly divisible by 22m.2, the second lowest 2.power with respect to the number of the Gaussian prime factors of D. As a corollary, we obtain a new series of non.congruent numbers whose prime factors can be arbitrarily many. Our result is in accord with the predictions of the conjecture of Birch and Swinnerton.Dyer.