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?

This is how I tried:

As we can see:

$$F(s)= \sum_m \mu(m) \sum_n \frac{e^{-n/2}\zeta^\prime (mns)}{n \zeta(mns)}$$

So we can see

$$F(1/2)=Res_{s=\frac{1}{2}} \frac{\int F(s)ds+(\frac{1}{4e}-\frac{1}{2√e})\ln(s-\frac{1}{2})}{(s-\frac{1}{2})²}$$

I think this is equal to zero under Riemann hypothesis. I need more insight into this. Any comments regarding this are welcome.

As we can see $\int F(s)ds$ has logarithmic singularity at s=1/2 so, the log term in the numerator is to cancel that singularity.

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:

On infinite sum containing logarithmic derivative of Zeta function and Möbius function: