$F\left(x,s\right)=\frac{\Gamma \left(1-s\right)}{{\left(2\pi \right)}^{1-s}}\left(e^{\pi i\left(1-s\right)/2}\zeta \left(1-s,x\right)+e^{\pi i\left(s-1\right)/2}\zeta \left(1-s,1-x\right)\right)$