It is unlikely that the following function has a closed form expression: $$f(t)=\sum_{n=0}^\infty\sum_{m=0}^\infty \frac{(-1)^{m+n}}{(2m+1)^3(2n+1)^3}\cos\left(\pi t \sqrt{(2m+1)^2+(2n+1)^2}\right)$$
It is clear that $|f(t)|\le \left(\sum_{n=0}^\infty\frac{1}{(2n+1)^3}\right)^2=\frac{49}{64}\zeta(3)^2$. My question is: What is $\sup_{t\ge 0} f(t)$? And what is $\limsup_{t\to\infty} f(t)$?