\input amstex 
\begin{align*} \int\limits_{\Gamma} f(z) \,dz &=& 2\pi i \left(\input amstex  \underset{ z=-1}\to{\operatorname{Res}} f(z)+\input amstex  \underset{ z=e^{i \frac{\pi}{3}}}\to{\operatorname{Res}} f(z)+\input amstex  \underset{ z=e^{-i \frac{\pi}{3}}}\to{\operatorname{Res}} f(z) \right)\\
&=& \displaystyle  2\pi i \left(\frac{-g^2(-1)}{3}+\frac{e^{i \frac{\pi}{3}}g^2(e^{i \frac{\pi}{3}})}{3e^{i \frac{2\pi}{3}}}+\frac{e^{-i \frac{\pi}{3}}g^2(e^{-i \frac{\pi}{3}})}{3e^{-i \frac{2\pi}{3}}}\right)\\
&=& \displaystyle 2\pi i \left(\frac{\pi^2}{3}-\frac{\pi^2}{27}e^{-i \frac{\pi}{3}}-\frac{25\pi^2}{27}e^{i \frac{\pi}{3}}  \right)\end{align*}