I was looking through the derivation of the Riemann functional equation, and I understand how to obtain the result
$$ \pi^{-\frac s2} \Gamma (\frac s2) \zeta(s) = \pi^{-\frac{1-s}{2}} \Gamma(\frac{1-s}{2}) \zeta(1-s)$$
which simplifies to
$$ \zeta(s) = \pi^{s - \frac12} \frac{\Gamma(\frac{1-s}{2})}{\Gamma (\frac s2)} \zeta(1-s) $$
What I can't seem to figure out is how to go from this functional equation to the more commonly presented functional equation
$$ \zeta(s) = 2^s \pi^{s-1} sin(\frac{\pi}{2}s) \Gamma(1-s) \zeta(1-s) $$
Can anyone explain this to me? I (probably a bit naively) plugged $ \frac{\Gamma(\frac{1-s}{2}}{\Gamma (\frac s2)} $ into WolframAlpha, and confirmed that this is equal to $ \frac{2^s sin(\frac{\pi}{2}s) \Gamma(1-s)}{\sqrt(\pi)}$ but after fiddling around with the Gamma function for a bit I couldn't figure out how to prove that.