zeta function-deriving reflection equation from functional equation From analytic continuation of zeta function, 
\begin{align}
   \zeta(s) = \frac{\pi^{\frac{s}{2}}}{\Gamma\left(\frac{s}{2}\right)} \left[ \frac{1}{s(s-1)} + \int_1^{\infty} \left( x^{\frac{s}{2}-1} + x^{-\frac{s}{2}-1} \right) \left(\frac{\theta(x) - 1}{2}\right) dx \right] \end{align}
From this I obtain a functional equation 
\begin{align}
\pi^{-\frac{s}{2}} \Gamma\left(\frac{s}{2}\right) \zeta(s) = \pi^{-\frac{1-s}{2}} \Gamma\left(\frac{1-s}{2}\right) \zeta(1-s)
\end{align}
My next step is to prove reflection functional equation 
\begin{align}
\zeta(1-s) = 2 (2\pi)^{-s} \cos\left(\frac{s\pi}{2}\right) \Gamma(s)\zeta(s)
\end{align}
Reordering a functional equation for $\zeta(s)$, I have 
\begin{align}
\zeta(1-s) = \pi^{\frac{1}{2}} \frac{\Gamma\left(\frac{s}{2}\right)}{\Gamma\left(\frac{1-s}{2}\right)} \zeta(s)
\end{align}
I have problem with showing above two equations are indeed same...

Use 
\begin{align}
  &\frac{\pi^{\frac{1}{2}}}{2^{s-1}} \Gamma (s) = \Gamma \left(\frac{s}{2}\right) \Gamma \left(\frac{s+1}{2}\right) \\
  & \Gamma\left(\frac{1+s}{2}\right) \Gamma\left(\frac{1-s}{2}\right) = \frac{\pi}{\cos\left(\frac{\pi s}{2}\right)} 
\end{align}
which comes from Legndre duplication formula and Euler's Reflection formula of gamma function.
Then I can prove the relation well!
