F ( x , s ) = ∑ n = 1 ∞ \expe 2 \cpi \iunit n x n s periodic-zeta 𝑥 𝑠 superscript subscript 𝑛 1 superscript \expe 2 \cpi \iunit 𝑛 𝑥 superscript 𝑛 𝑠 {\displaystyle{\displaystyle{\displaystyle F\left(x,s\right)=\sum_{n=1}^{% \infty}\frac{\expe^{2\cpi\iunit nx}}{n^{s}}}}} {\displaystyle \PeriodicZeta@{x}{s} = \sum_{n=1}^\infty \frac{\expe^{2 \cpi \iunit n x}}{n^s} }
F ( x , s ) = Γ ( 1 - s ) ( 2 \cpi ) 1 - s ( \expe \cpi \iunit ( 1 - s ) / 2 ζ ( 1 - s , x ) + \expe \cpi \iunit ( s - 1 ) / 2 ζ ( 1 - s , 1 - x ) ) periodic-zeta 𝑥 𝑠 Euler-Gamma 1 𝑠 superscript 2 \cpi 1 𝑠 superscript \expe \cpi \iunit 1 𝑠 2 Hurwitz-zeta 1 𝑠 𝑥 superscript \expe \cpi \iunit 𝑠 1 2 Hurwitz-zeta 1 𝑠 1 𝑥 {\displaystyle{\displaystyle{\displaystyle F\left(x,s\right)=\frac{\Gamma\left% (1-s\right)}{(2\cpi)^{1-s}}\*\left(\expe^{\cpi\iunit(1-s)/2}\zeta\left(1-s,x% \right)+\expe^{\cpi\iunit(s-1)/2}\zeta\left(1-s,1-x\right)\right)}}} {\displaystyle \PeriodicZeta@{x}{s} = \frac{\EulerGamma@{1-s}}{(2 \cpi)^{1-s}} \* \left( \expe^{\cpi \iunit (1-s)/2} \HurwitzZeta@{1-s}{x} + \expe^{\cpi \iunit (s-1)/2} \HurwitzZeta@{1-s}{1-x} \right) }
ζ ( 1 - s , x ) = Γ ( s ) ( 2 \cpi ) s ( \expe - \cpi \iunit s / 2 F ( x , s ) + \expe \cpi \iunit s / 2 F ( - x , s ) ) Hurwitz-zeta 1 𝑠 𝑥 Euler-Gamma 𝑠 superscript 2 \cpi 𝑠 superscript \expe \cpi \iunit 𝑠 2 periodic-zeta 𝑥 𝑠 superscript \expe \cpi \iunit 𝑠 2 periodic-zeta 𝑥 𝑠 {\displaystyle{\displaystyle{\displaystyle\zeta\left(1-s,x\right)=\frac{\Gamma% \left(s\right)}{(2\cpi)^{s}}\left(\expe^{-\cpi\iunit s/2}F\left(x,s\right)+% \expe^{\cpi\iunit s/2}F\left(-x,s\right)\right)}}} {\displaystyle \HurwitzZeta@{1-s}{x} = \frac{\EulerGamma@{s}}{(2 \cpi)^s} \left( \expe^{-\cpi \iunit s/2} \PeriodicZeta@{x}{s} + \expe^{\cpi \iunit s/2} \PeriodicZeta@{-x}{s} \right) }