In skimming through Narkiewicz "The Development of Prime Number Theory", one sees a reference on p. 155 (footnote 38) to a certain R. Lipschitz, who in Crelle in 1857 "studied the series $\sum_{n=1}^\infty\exp(nui)n^{-\sigma}$ for real values of $\sigma$." I checked the reference; Lipschitz was indeed using $\sigma$.
Lipschitz is referred to several times in this section of Narkiewicz for later work on functional equations of various $L$-functions.