Is it true that $$ \zeta(s)\\\equiv\frac{e^{(\log(2\pi)-1-\frac{1}{2}\gamma)s}}{2(s-1)\Gamma(1+\frac{1}{2}s)}\prod_{n=1}^{\infty}\bigg(1-\frac{s}{\rho_{n}}\bigg)e^{s/\rho_{n}}\\\equiv\frac{1}{s-1}+\gamma+\sum_{n=1}^{\infty}\frac{-1^{n}}{n!}\gamma_{n}(s-1)^{n}\\\equiv\prod_{n=1}^{\infty}\frac{1}{1-p_{n}^{-s}}$$ (where $p_{n}\text{ is the }nth \text{ prime },\ \rho_{n}\text{ is the }nth \text{ zeta zero },\ \text{ and }\gamma \text{ is Euler's constant}$)
over the whole of the complex plane? If not, which is not analytically continuable, and why?