Let $\alpha\in\mathbb R -\mathbb Z$, and let $z\in\mathbb C$. Is there any closed form expression for either one of the two infinite products: $$\prod_{n\in\mathbb Z} \frac{z^2 +(n+\alpha)^2}{(n+\alpha)^2}, \qquad \textrm{or}\qquad \prod_{n\in\mathbb N} \frac{z^2 +(n+\alpha)^2}{n^2} \:?$$ Note that both infinite products are well-defined.
If $\alpha=0$, I know that we have $$\prod_{n\in\mathbb N} \frac{z^2+n^2}{n^2} = \frac{\sinh(\pi z)}{\pi z}.$$ I want a generalization of this formula.
$$P_p=\prod_{n=1}^p \frac{z^2 +(n+\alpha)^2}{(n+\alpha)^2}=\prod_{n=1}^p \frac{\big[z+(n+\alpha)i\big]\,\big[z-(n+\alpha)i\big]}{(n+\alpha)^2}$$
Using Pochhammer symbols, $$P_p=\frac{(\alpha +1-i z)_p \,\,(\alpha +1+iz)_p}{\big[(\alpha +1)_p\big]{}^2}$$ $$\log(P_p)=\log \left(\frac{\Gamma (\alpha +1)^2}{\Gamma (\alpha +1-iz)\,\, \Gamma ( \alpha +1+iz)}\right)-\frac{z^2}{p}+\frac{(2 \alpha +1) z^2}{2 p^2}+O\left(\frac{1}{p^3}\right)$$ $$P_\infty=\frac{\Gamma (\alpha +1)^2}{\Gamma (\alpha +1-iz)\,\, \Gamma ( \alpha +1+iz)}$$
For some values of $\alpha$, the infinite product $$Q_\alpha=\prod_{n=1}^\infty \frac{z^2 +(n+\alpha)^2}{(n+\alpha)^2}$$ seems to be quite nice.
$$Q_0=\frac{\sinh (\pi z)}{\pi z}\quad\quad Q_1=\frac{\sinh (\pi z)}{\pi z(z^2+1) }\quad\quad Q_2=\frac{4\sinh (\pi z)}{\pi z \left(z^2+1\right) \left(z^2+4\right) }$$ $$Q_{\frac 12}=\frac{\cosh (\pi z)}{4 z^2+1}\quad\quad Q_{\frac 32}=\frac{9 \cosh (\pi z)}{\left(4 z^2+1\right) \left(4 z^2+9\right)}$$