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:
$\hat{\displaystyle\prod_{p\equiv 1 ~(\text{mod}~4)}}p = \dots$
$\hat{\displaystyle\prod_{p,n}}{~p^{n}}~ = \dots$
$\hat{\displaystyle\prod_{p,n}}{~p^{2^n}}~ = \dots$
$\hat{\displaystyle\prod_{\pi(p) \equiv 1 ~(\text{mod}~2)}}{p} = \dots$
$\hat{\displaystyle\prod_p}{(p+1)} = \dots$