Consider the following even meromorphic doubly periodic function with poles at the gaussian integer lattice.

$H(z) = \prod_{n \in \mathbb{Z}} {1 \over{ 1 - {1 \over{\cosh^2\left(2\pi\left(z-n\right)\right)}}}}$

Computational evidence suggests special values for: $H({1\over4}) = H({3\over4}) = 0$

and rather amazingly

$Real \left( H(z) \right) = H\left({{1+i}\over 2}\right) = 0.847201266746891$

to remain constant on the of the unit square diagonals.

How would one go proving this and/or finding an exact analytic formula for this constant ?