\[F\left(3a,\tfrac{1}{3}+a;\tfrac{2}{3}+2a;{\mathrm{e}}^{\ifrac{\mathrm{i}\pi}{3%
}}\right)=\sqrt{\pi}{\mathrm{e}}^{\ifrac{\mathrm{i}\pi a}{2}}\left(\frac{16}{2%
7}\right)^{(3a+1)/6}\frac{\Gamma\left(\frac{5}{6}+a\right)}{\Gamma\left(\frac{%
2}{3}+a\right)\Gamma\left(\frac{2}{3}\right)},\]