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Questions about generalizations of the Riemann Zeta function of arithmetic interest whose definition relies on meromorphic continuation of special kinds of Dirichlet series, such as Dirichlet L-functions, Artin L-functions, elements of the Selberg class, automorphic L-functions, Shimizu L-functions, p-adic L-functions, etc.

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Complex L-functions for Hermitian modular forms?

To define an "L-function" for an automorphic representation $\pi$ on a group $G$ you usually have to make a choice of a representation $\rho$ of the dual group. Only then can you define $L(\pi,\rho,s) …
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