Let D = p1 … pm, where p1, …, pm are distinct rational primes ≡ 1(mod 8), and m is any positive integer. In this paper, we give a simple combinatorial criterion for the value of the Hecke L-function of the congruent elliptic curve ED2 : y2 = x3 − D2x at s = 1, divided by the period ω defined below, to be exactly divisible by 4m. As a corollary, we obtain a series of non-congruent numbers whose number of prime factors tends to infinity, and for which the corresponding elliptic curves have non-trivial 2-part of Tate–Shafarevich group, which greatly generalizes a result of Razar [8]. Our result is in accord with the predictions of the conjecture of Birch and Swinnerton-Dyer.