What is the value of this prime product formula:
$$\prod_{p=1,2\pmod5}\left(1-\frac{1}{p}\right)^{-1}\prod_{q=3,4\pmod5}\left(1+\frac{1}{q}\right)^{-1}$$
I tried to translate into sums over certain mod5 reciprocal numbers following certain patterns but unsuccessfully. Following same approach, we can get closed-forms for corresponding expressions mod $3$ and $4$ that are related to $\pi$.
Thanks in advance