TITLE: Are there refuted analogues of the Riemann hypothesis?
QUESTION [56 upvotes]: The classical Riemann Hypothesis has famous analogues for function fields and finite fields which have been proved.  It has by now very many analogues, many of them still open.  Are there important analogues that are now known to be false?

REPLY [12 votes]: It is some time ago that I read it, so I hope I recall the details correctly but:
in Eisenstein series and the Riemann zeta function (Automorphic Forms, Representation Theory and Arithmetic, Springer-Verlag, Berlin-Heidelberg-New York (1981) 275-301)
Don Zagier shows that the RH is equivalent to the unitarity of a certain (non-admissible) $SL(2, \mathbb{R})$ representation. Next he gives a second construction of this representation in terms of the adeles over $\mathbb{Q}$. Then a rather surprising punch line follows: he applies the same construction to the adeles over $\mathbb{Q}(\sqrt{2})$, thus obtaining a similar represenation and shows that this representation is not unitary (by exhibiting an invariant subspace that is not a direct summand).
Although it is more a variant of an equivalent reformulation of the RH that is shown false than a variant of the RH itself, at least it goes to show, as Zagier points out, that any proof of the RH should exploit some very special properties of the rationals not shared by just any global field.  
EDIT: I just found an online version of the paper and it turns out I remembered it wrongly. Apparently the unitarity of the mentioned representation is not equivalent to RH but strictly stronger. In particular it would also imply that the zeroes of zeta are simple (which is believed to be even harder than RH according to the answer to this MO question: Are the nontrivial zeros of the Riemann zeta simple?.) Apparently this harder conjecture is known to be false over fields other than $\mathbb{Q}$ and it is this fact that is used to construct the counterexample to the unitarity of the analogous representations coming from other fields (also, to my surprise, $\mathbb{Q}(\sqrt{2})$ is mentioned nowhere explicitly). So not an answer to the original question after all, I'm sorry. Still the article is quite interesting.