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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.

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The answer is either yes or no, depending on how stringently you interpret your various requirements. You should look at the discussion of the Selberg class of functions, which is Selberg's conjectural characterization of functions satisfying the Riemann Hypothesis. In particular, if you read the comments on the definition in the wikipedia entry, you will get a sense of why those are the precise conditions on a "$\zeta$-type function" which are needed to guarantee RH.

As a concrete (counter-)example, consider the function $$\eta(s) = 1 - 2^{-s} + 3^{-s} - 4^{-s} + \cdots,$$ sometimes called the Dirichlet $\eta$-function. It admits a functional equation and Euler product, but does not satisfy RH. It is not in the Selberg class because although it admits an Euler product, its Euler factors do not satisfy the correct conditions. (This is discussed in the wikipedia entry on the Selberg class.)

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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

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