In skimming through Narkiewicz "The Development of Prime Number Theory", one sees a reference on p. 155 (footnote 38) to a R. Lipschitz, who in Crelle volume 54 in 1857 "studied the series $\sum_{n=1}^\infty\exp(nui)n^{-\sigma}$ for real values of $\sigma$." I checked the reference, which is here (there are two papers by Lipschitz in this volume of Crelle, and it's the second one that's relevant); 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.
