I=-\frac{1}{4\pi} \Im \left[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)\right]=-\frac{1}{4\pi} \left[ \frac{2\pi^3}{3} -\frac{2\pi^3}{27} \cos \frac{\pi}{3}-\frac{50 \pi^3}{27} \cos \frac{\pi}{3} \right]=-\frac{1}{4\pi} \left[\frac{2\pi^3}{3} -\frac{\pi^3}{27} -\frac{25 \pi^3}{27}  \right]=\frac{2\pi^2}{27}