Closed form for $\frac{1}{2 \pi z}\sum _{n=-\infty }^{\infty } \sum _{m=-\infty }^{\infty } e^{-z \sqrt{m^2+n^2}} \cos (m y) \cos (n x)$

Question

I looking for a closed form of this double sum, which is not tractable with mathematica. $$\frac{1}{2 \pi z}\sum _{n=-\infty }^{\infty } \sum _{m=-\infty }^{\infty } e^{-z \sqrt{m^2+n^2}} \cos (m y) \cos (n x)$$

Any suggestions or ideas are welcomed.