Reading some notes I have taken during class, I am unable to see the connection between these lines: $g(z)=\prod_{k\geqslant 0}(2z\lambda_k + 1)$, where $\lambda_k=4/((2k+1)^2 \pi^2)$
And
The zeroes of $h(z)=g(z^2)$ are $(i/\sqrt{2\lambda_k})$ (ok, why not) which implies $g(z^2)=\sinh(\sqrt{2z})$ (how?).
Appreciate any hints.