Generalized ZETA function

[zetafunction]

given the function Z(s)= \prod _{k=0}^{\infty}\zeta (s+k) with \zeta (s) being the Riemann Zeta function

the idea is if ALL the roots have real part (i mean Riemann Hypothesis) is correct, then what would happen with the roots of Z(s) ??

what would be the Functional equation relating Z(1-s) and Z(s) ¿¿ from the definition of Riemann functional equation

[Petek]

For which values of s does Z(s) exist? A necessary condition for an infinite product to converge is that the individual terms converge to 1. Thus, it must be true that

\zeta(s + k) \rightarrow 1 as k \rightarrow \infty

It's not clear to me for which values of s this holds.

Petek

[g_edgar]

True for Re s > 1, right? And therefore true for all s ...

[Petek]

True for Re s > 1, right? And therefore true for all s ...

Yes, that's right. Thanks!

Petek