J=\frac{1}{4\pi^2} \Re \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}{27} \sin \frac{\pi}{3}+\frac{50 \pi^3}{27} \sin \frac{\pi}{3} \right]=\frac{1}{4\pi^2} \left[ -\frac{\sqrt{3}\pi^3}{27}+\frac{25\sqrt{3}\pi^3}{27}\right]=\frac{2\sqrt{3}}{9}\pi