Assume $s,a \in \mathbb{C}, a \pm in \ne 0$.


The following infinite product nicely converges and can be expressed in a closed form:


$$\displaystyle \prod\_{n=1}^\infty \left(1- \frac{s}{a+i n} \right) \left(1- \frac{s}{{a- i n}} \right) = {\frac {a\sinh \left( \pi \left( a-s \right) \right) }{ \left( a-s
 \right) \sinh \left( \pi a \right) }}$$


The individual factors however diverge, so I tried to exchange the sub factors for each $n$ and found that:


$$\displaystyle \prod\_{n=1}^\infty \left(1- \frac{s}{a+ (-1)^n i n} \right)$$ 


and


$$\displaystyle \prod\_{n=1}^\infty \left(1- \frac{s}{a+ (-1)^{n+1} i n} \right)$$ 


actually do (slowly but surely) converge and when multiplied together yield the closed form above.


Is there a closed form for these two individual factors?


**EDIT (and follow up question):**


Many thanks to Carlo for answering the question so quickly. 


A (maybe too) provocative follow up question deals with a similar product that also has a closed form (when assuming RH is true):


$$\prod\_{n=1}^\infty \left(1- \frac{s}{a + i \Im(\rho\_n)} \right) \left(1- \frac{s}{{a - i \Im(\rho\_n)}} \right) = \frac{\xi\_{rie}(\frac12 - a+s)}{\xi\_{rie}(\frac12 - a)}$$


so runs through the non-trivial zeros $\rho\_n$ with:$\xi\_{rie}(s)= \frac12 s(s-1) \pi^{-\frac{s}{2}} \Gamma(\frac{s}{2}) \zeta(s)$


Again, the individual factors diverge, but after exchanging the sub factors for each $n$, I again found that:


$$\displaystyle \prod\_{n=1}^\infty \left(1- \frac{s}{a+ (-1)^n i \Im(\rho\_n)} \right)$$


and


$$\displaystyle \prod\_{n=1}^\infty \left(1- \frac{s}{a+ (-1)^{n+1} i \Im(\rho\_n)} \right)$$


actually do converge.


A closed form for these would obviously be quite spectacular... Could it exist? 


P.S.:


In another context, I asked about the similarity between these products here: [closed forms infinite products](https://mathoverflow.net/questions/122582/is-there-a-connection-between-the-closed-forms-of-these-two-infinite-products)



Does there exist a closed form for the factors of this infinite product ?
$\displaystyle \prod\_{n=1}^\infty \left(1- \frac{s}{a+ (-1)^n i n} \right)=\frac{\Gamma \left(\frac{1}{2}+\frac{i a}{2}\right) \Gamma \left(1-\frac{i a}{2}\right)}{\Gamma \left(\frac{1}{2}-\frac{1}{2} i (s-a)\right) \Gamma \left(1+\frac{1}{2} i (s-a)\right)}$


$\displaystyle \prod\_{n=1}^\infty \left(1- \frac{s}{a+ (-1)^{n+1} i n} \right)=\frac{\Gamma \left(1+\frac{i a}{2}\right) \Gamma \left(\frac{1}{2}-\frac{i a}{2}\right)}{\Gamma \left(1-\frac{1}{2} i (s-a)\right) \Gamma \left(\frac{1}{2}+\frac{1}{2} i (s-a)\right)}$


