It is becoming increasingly clear that the expression $~\displaystyle\prod_{n\in\mathbf M}\left[1-\frac{z^s}{(n-a)^s}\right],~$ with $~|\mathbf M|=\aleph_0,~$

$a\not\in\mathbf M,~$ and $~\Re(s)>1,~$ lends itself as a natural generalization of all four classes of functions

mentioned in the title, since

• for natural values of n and s, it yields a (finite) product of $\Gamma$ functions, which, in certain cases (depending on the parity of s, and on whether a is either an integer or a half-integer), can be further simplified to a (finite) product of trigonometric and/or hyperbolic functions.

• for prime values of n, with $a=0,$ and $|z|=1,$ it yields a (finite) product of $\zeta$ functions.

My actual question would be what exactly happens in the former case for $\color{blue}{s\in\mathbb R\setminus\mathbb N},$ and in the latter for $\color{blue}{|z|\not\in\{0,1\}}.~$ But just in case answering this might prove too vast of an undertaking, I've decided it might be best to try and simplify things a bit by splitting it into two distinct sub-questions,

asking specifically for the evaluation of $~\displaystyle\prod_{n\in\mathbb N^\star}\left(1+n^{-3/2}\right),~$ and $~\displaystyle\prod_{p\in\mathbb P}\left(1+\frac{2^s}{p^s}\right).~$ The very

first thought crossing my mind is whether any of the two (or perhaps something very similar to them) has ever been mentioned in the literature. The other one would be to inquire for acceleration methods pertaining to the (rather thorny) issue of their numerical computation, since all mathematical software I can think of (Mathematica, Maple, PARI/GP) experience significant hurdles whilst trying to determine even the first few digits of their decimal representation. (Thus, all I know so far about the first product, for instance, is that it is approximately equal to 9.20). Any help would be deeply appreciated !