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