I am trying to understand a paper by by A. Booker on poles of Artin $L$-functions where in one of the lemmas he uses the following identity, derived using Stirling's formula: $$ \frac{\Gamma(s/2)^2}{2^{-s}\Gamma(s-1/2)} = \sqrt{8\pi}\left( 1+\frac{c_1}{s}+\ldots \frac{c_n}{s^n}+O\left(\frac{1}{s^{n+1}}\right)\right),\qquad \Re s\geq 1 $$ (for certain numbers $c_k$).

I don't quite understand how they got this, can anyone help?