Formula:8432

Digital Library of Mathematical Functions ID 25.13.E3

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

Constraint(s)

${\displaystyle{\displaystyle\Re s>0}}$
${\displaystyle{\displaystyle 0<x<1}}$
${\displaystyle{\displaystyle\Re s>0}}$

Symbols List

• ${\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}$: gamma function
• ${\displaystyle{\displaystyle\zeta\left(\NVar{s},\NVar{a}\right)}}$: Hurwitz zeta function
• ${\displaystyle{\displaystyle\pi}}$: the ratio of the circumference of a circle to its diameter
• ${\displaystyle{\displaystyle\mathrm{e}}}$: base of natural logarithm
• ${\displaystyle{\displaystyle\mathrm{i}}}$: imaginary unit
• ${\displaystyle{\displaystyle F\left(\NVar{x},\NVar{s}\right)}}$: periodic zeta function
• ${\displaystyle{\displaystyle\Re}}$: real part
• ${\displaystyle{\displaystyle x}}$: real variable
• ${\displaystyle{\displaystyle s}}$: complex variable
 

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