Theorem 3.4 (Functional equation for zeta). Assuming that:

  • ξ(s)=12s(s−1)π−s∕2Γ(ss)ζ(s) for s∈ℂ

Then ξ is an entire function and ξ(s)=ξ(1−s) for s∈ℂ. Hence
π−s∕2Γ(s2)ζ(s)=π−1−s2Γ(1−s2)ζ(1−s)

for s∈ℂ∖{0,1}.