I know this forum is for professional Mathematicians. I didn't know if the following question is fit or not fit for the site. If not, feel free to delete it.

>Consider the modified Euler product as follows:
>
>$$F(s) = \prod_{p} \left( 1 - \frac{c}{p^s} \right)^{-\ln(p)}$$
>
>Here $c$ is a constant 

My questions are 

1. Is there a compact representation for this product?

2. What are some non-trivial  properties of this product?


3. Value of the regularized sum:

$$-\sum_{p}\ln\left( 1 - \frac{1}{(ep)^{1/2}} \right){\ln(p)}$$

Help me understand the analytic continuation of this function.


Or  consider

$$f(s) = \sum_{p}\left( 1 - \frac{1}{(ep)^{1/2}} \right)^{s\ln(p)}$$

$p$ belongs to set of primes


We can find f'(x)'s analytic continuation for some region containing s=0.


(This is the main motivation behind the question)

Edit:  Now by using the answer by R.Furman how can we calculate the product's value 'uniquely' at 1/2?






I think this is equal to zero under Riemann hypothesis. Also I think  if its equal to zero then RH is true . So, I  need more insight into this. Any comments regarding this are welcome.

Edit:

We can also have a generalized conjecture:


>$$ \prod_{p} \left( 1 - \frac{1}{e^{\alpha}p^{\beta}} \right)^{-\ln(p)}=\left|\frac{(\zeta({\alpha}+{\beta})({\alpha}+{\beta}-1)}{{\alpha}-{\beta}+1}\right|$$

 Here,     $ {\alpha} >0$  ${\beta}$ belongs to $[1/2,1)$ s.t.
     ${\alpha}+{\beta} \notequalto 1$ 









References:

[1]    Germund Dahlquist ; "On the analytic continuation of Eulerian products"

[2] https://mathworld.wolfram.com/PrimeProducts.html ( and all the references therein)

[3] https://en.m.wikipedia.org/wiki/Euler_product

[4] Kimoto , Wakayama "Remarks on Zeta Regularized Products"

Progress post:

https://mathoverflow.net/q/386245/174361

Recently I found a similar post while surfing through the site:

https://mathoverflow.net/q/275706/174361

If anyone could give answer in the context of the above post it will be very nice.