Euler product on the critical line

Question

can the Euler product be defined on the critical line $ 1/2+it $ ??

i mean the Euler product for the Riemann Zeta function $$ \zeta(s)= \prod _{n}(1-p^{-s})^{-1} $$

for example from the representation of the Dirichlet generating function for the Rieman zeta would we have that ??

$$ \prod _{n\le x}(1-p^{-0.5-it})^{-1} =\exp\left(\sum_{n \le x}\frac{\Lambda(n)}{n^{0.5+it}}\right)$$

Answer 1

No. See https://mathoverflow.net/questions/63714/ ${ }$ ${ }$ ${ }$ ${ }$