In [1], Lin Weng shows that the Riemann hypothesis (RH) holds for certain linear combinations of shifted Riemann zetas. I would like to know whether these are all examples for which the RH is known.

To be precise, we say that a Dirichlet series $$ D(s)=\sum_{n=1}^\infty \frac{a_n}{n^s} $$ satisfies the generalized Riemann hyporthesis (genRH), if it extends to a meromorphic function on $\mathbb C$ all of whose poles and zeroes lie in the union of a finite number of vertical lines $c+i\mathbb R$ and the real line. Are there any known examples, other than those in [1], for which the genRH has been proven?

[1] Symmetries and the Riemann hypothesis. Algebraic and arithmetic structures of moduli spaces (Sapporo 2007), 173–223, Adv. Stud. Pure Math., 58, Math. Soc. Japan, Tokyo, 2010.

In [1], Lin Weng shows that the Riemann hypothesis (RH) holds for certain linear combinations of shifted completed Riemann zetas. Further, Deligne's result on the Riemann hypothesis for function fields gives a RH for the latter. I would like to know whether these are all examples for which the RH is known.

To be precise, we say that a meromorphic function on $\mathbb C$ satisfies the generalized Riemann hyporthesis (genRH), if all of its poles and zeroes lie in the union of a finite number of vertical lines $c+i\mathbb R$ and the real line. Are there any known examples, other than those in [1], or those derived from Deligne's work, for which the genRH has been proven? Of particular interest would be a function which can be written as a Dirichlet series for $\Re(s)>>0$.

[1] Symmetries and the Riemann hypothesis. Algebraic and arithmetic structures of moduli spaces (Sapporo 2007), 173–223, Adv. Stud. Pure Math., 58, Math. Soc. Japan, Tokyo, 2010.