F â¡ ( x , s ) = â n = 1 â \expe 2 ⢠\cpi ⢠\iunit ⢠n ⢠x n s periodic-zeta ð¥ ð superscript subscript ð 1 superscript \expe 2 \cpi \iunit ð ð¥ superscript ð ð {\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 â¡ ( x , s ) = Î â¡ ( 1 - s ) ( 2 ⢠\cpi ) 1 - s ⢠( \expe \cpi ⢠\iunit ⢠( 1 - s ) / 2 ⢠ζ â¡ ( 1 - s , x ) + \expe \cpi ⢠\iunit ⢠( s - 1 ) / 2 ⢠ζ â¡ ( 1 - s , 1 - x ) ) periodic-zeta ð¥ ð Euler-Gamma 1 ð superscript 2 \cpi 1 ð superscript \expe \cpi \iunit 1 ð 2 Hurwitz-zeta 1 ð ð¥ superscript \expe \cpi \iunit ð 1 2 Hurwitz-zeta 1 ð 1 ð¥ {\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 â¡ ( 1 - s , x ) = Î â¡ ( s ) ( 2 ⢠\cpi ) s ⢠( \expe - \cpi ⢠\iunit ⢠s / 2 ⢠F â¡ ( x , s ) + \expe \cpi ⢠\iunit ⢠s / 2 ⢠F â¡ ( - x , s ) ) Hurwitz-zeta 1 ð ð¥ Euler-Gamma ð superscript 2 \cpi ð superscript \expe \cpi \iunit ð 2 periodic-zeta ð¥ ð superscript \expe \cpi \iunit ð 2 periodic-zeta ð¥ ð {\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) }