TITLE: Are there examples that suggest the Riemann Hypothesis might be false?
QUESTION [29 upvotes]: 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.

REPLY [11 votes]: There is the de-Bruijn Newman Constant $\Lambda $ . RH is equivalent that $\Lambda  \leq 0$. Up to now it is known  $-2.7 \cdot 10^{-9} < \Lambda  \leq 1/2$. See http://www.dtc.umn.edu/~odlyzko/doc/debruijn.newman.pdf