Let D=π1..., πn, where π1, ... πn are distinct Gaussian primes ≡ (mod 4) and n is any positive integer. In this paper, we prove that the value of the Hecke L-function attached to the elliptic curve ED2: y2=x3−D2x at s=1, divided by the period ω defined below, is always divisible by 2n−1. Moreover, we give a simple combinatorial criterion for this value to be exactly divisible by 2n−1. Our results are in accord with the predictions of the conjecture of Birch and Swinnerton-Dyer, and, when D is rational, enable us to prove the conjecture of Birch and Swinnerton-Dyer for ED2 when the value at s=1 is exactly divisible by 2n−1.