I took some notes on the historical use of zeta functions in number theory here Why the zeta function? but from looking up the dates I didn't realize something:
I remember reading that its not called the Euler zeta function because he never analytically continued it - even though Euler did lots with it, including studying primes.
But it seems that Dirichlet did, and must he not have also found the functional equation? So why did Riemann think to study the zeta function? wasn't he inspired by Dirichlet's work as well as Eulers?