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 [post](https://mathoverflow.net/questions/91763/special-values-of-a-doubly-periodic-meromorphic-function)



Elliptic function with constant real part on the unit square diagonals?
Well, again you seem to have a constant multiple of the Weierstrass P-function, plus a constant. There is a formula in this case for P((1 + i)z), not just for P(2z) as there is for any period lattice. The tilting line with constant real part is clearly related. The fact that the real part is constant will be an aspect of the Schwarz reflection principle, which tells one the condition that a function is real on the real axis (<http://en.wikipedia.org/wiki/Schwarz_reflection_principle>). 


