As a meromorphic doubly-periodic function, it is an elliptic function in the classical sense. You didn't mention the fact that it is an even function, but that also helps. The poles are double, so that up to addition of a constant this must be a constant multiple of the Weierstrass P-function for the lattice of Gaussian integers. Since the P-function in this case is 0 at a known point, your third value gives the additive constant as (presumably) 0.5.
Everything about this function is standard in the theory of complex multiplication. However you came up with it, the formulae are probably written down somewhere.