Davenport (Chapter 9 in Multiplicative Number Theory) claims that the functional equation for Dirichlet L-functions was first given by Hurwitz in 1882 (Werke I, pp.72-88), though only for quadratic characters. The proof uses what we now call the Hurwitz zeta function.
I was told just yesterday that some people refer to the Riemann Hypothesis for Dirichlet L-functions as the Piltz Hypothesis. This is confirmed in the wikipedia article.