What infinite prime products have $\zeta$-regularized values?

MathWorld gives just a few, for example:

$\hat{\displaystyle\prod_p}{p} = 4 \pi^2$

$\hat{\displaystyle\prod_p}{(p^2 - 1)} = \frac{(2 \pi)^4}{\zeta(2)} = 96\pi^2$

Do the above statements depend on the Riemann hypothesis, or are they otherwise conditional?

Are the values of other $\zeta$-regularized products involving primes known?

For example, what about these:

$\hat{\displaystyle\prod_{p\equiv 1 ~(\text{mod}~4)}}p = \dots$

$\hat{\displaystyle\prod_{p,n}}{~p^{n}}~ = \dots$

$\hat{\displaystyle\prod_{p,n}}{~p^{2^n}}~ = \dots$

$\hat{\displaystyle\prod_{\pi(p) \equiv 1 ~(\text{mod}~2)}}{p} = \dots$

$\hat{\displaystyle\prod_p}{(p+1)} = \dots$

  • 1
    $\begingroup$ Compute the logarithmic derivative, see Smirnov. He obtains $\hat{\displaystyle\prod_p}{p} = \pi e^{2+\gamma-\eta}$, hmmm. $\endgroup$ – Dietrich Burde Feb 1 '18 at 9:34
  • $\begingroup$ I had recently skimmed this already, part of why I am asking this is that the assumptions leading to $\eta = \gamma + 2 - \log{4 \pi}$ are not explained clearly. According to the abstract the value depends on the location of the zeroes, and other sources make no mention of this. $\endgroup$ – Dan Brumleve Feb 1 '18 at 9:41

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