Are there examples that might suggest the Riemann hypothesis is false?
I mean, is there a zeta function $ \zeta (s,X) $ for some mathematical object $X$ with the properties
$ \zeta (1-s,X) $ and $ \zeta (s,X)$ are related by a functional equation.
$\zeta (s,X) $ can be expanded into an Euler product $ \zeta (s,X)= \prod _{i}(1-N(i)^{-s})^{-1}$.
the zeta function $\zeta (s,X) $ has zeroes of the form $ a+bi$ with $b\ne0$ for $a$ different from $\frac 12$.
That is, a zeta function with similar properties to the Riemann zeta but with zeroes off the critical line.