The L-function is expected to have an analytic continuation as a meromorphic function of all complex , and satisfy a functional equation
where the factor is a product of "local constants"
almost all of which are 1.
General linear groups
Godement & Jacquet (1972) constructed the automorphic L-functions for general linear groups with r the standard representation (so-called standard L-functions) and verified analytic continuation and the functional equation, by using a generalization of the method in Tate's thesis. Ubiquitous in the Langlands Program are Rankin-Selberg products of representations of GL(m) and GL(n). The resulting Rankin-Selberg L-functions satisfy a number of analytic properties, their functional equation being first proved via the Langlands–Shahidi method.
In general, the Langlands functoriality conjectures imply that automorphic L-functions of a connected reductive group are equal to products of automorphic L-functions of general linear groups. A proof of Langlands functoriality would also lead towards a thorough understanding of the analytic properties of automorphic L-functions.
Jacquet, H.; Piatetski-Shapiro, I. I.; Shalika, J. A. (1983), "Rankin-Selberg Convolutions", Amer. J. Math., 105 (2): 367–464, doi:10.2307/2374264, JSTOR2374264