Riemann's thinking on symmetrizing the zeta functional equation

Although Riemann was the first to prove the functional equation for complex $s$, the ideas certainly predate Riemann. Euler knew something equivalent to the functional equation for integer values of $s$, via Abel summation where the series does not converge. Weil's 1975 paper "Two lectures on number theory, past and present" (in his collected works vol. 3) says that before Riemann, both Schlomilch and Malmquist published the functional equation for the Dirichlet $L$-series attached to the nontrivial character modulo $4$. This paper is a good place to begin for thinking about how Riemann's ideas evolved.