I found this function:
$$ \zeta(\alpha+i\beta)*\zeta(\alpha-i\beta)=\prod \frac{1}{1-\frac{2cos(\beta ln{p})}{p^\alpha} + \frac{1}{p^{2\alpha}}}=\Re(\zeta(\alpha+i\beta))^2 +\Im(\zeta(\alpha+i\beta))^2 $$
so its a real function with values such as $(2+i0) -> (\frac{\pi^2}{6})^2$ and so on and so forth. But I'm not too sure how to extend it because it seems like its only convergent in the region with real > 1 as usual. I don't see how to extend it. I tried alternating sign like eta but to no avail. Anybody knows how to extend it to below 1?
Thank you