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

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

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

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?

Similar interrogations also arise in this <a href="http://mathoverflow.net/questions/91763/special-values-of-a-doubly-periodic-meromorphic-function">post</a>