# §25.4 Reflection Formulas

For $s\neq 0,1$,

 25.4.1 $\zeta\left(1-s\right)=2(2\pi)^{-s}\cos\left(\tfrac{1}{2}\pi s\right)\Gamma% \left(s\right)\zeta\left(s\right),$
 25.4.2 $\zeta\left(s\right)=2(2\pi)^{s-1}\sin\left(\tfrac{1}{2}\pi s\right)\Gamma\left% (1-s\right)\zeta\left(1-s\right).$

Equivalently,

 25.4.3 $\xi\left(s\right)=\xi\left(1-s\right),$ ⓘ Symbols: $\xi\left(\NVar{s}\right)$: Riemann’s $\xi$-function and $s$: complex variable Permalink: http://dlmf.nist.gov/25.4.E3 Encodings: TeX, pMML, png See also: Annotations for 25.4 and 25

where $\xi\left(s\right)$ is Riemann’s $\xi$-function, defined by:

 25.4.4 $\xi\left(s\right)=\tfrac{1}{2}s(s-1)\Gamma\left(\tfrac{1}{2}s\right)\pi^{-s/2}% \zeta\left(s\right).$ ⓘ Defines: $\xi\left(\NVar{s}\right)$: Riemann’s $\xi$-function Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\zeta\left(\NVar{s}\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter and $s$: complex variable Permalink: http://dlmf.nist.gov/25.4.E4 Encodings: TeX, pMML, png See also: Annotations for 25.4 and 25

For $s\neq 0,1$ and $k=1,2,3,\dots$,

 25.4.5 $(-1)^{k}{\zeta^{(k)}}\left(1-s\right)=\frac{2}{(2\pi)^{s}}\sum_{m=0}^{k}\sum_{% r=0}^{m}\genfrac{(}{)}{0.0pt}{}{k}{m}\genfrac{(}{)}{0.0pt}{}{m}{r}\left(\Re(c^% {k-m})\cos\left(\tfrac{1}{2}\pi s\right)+\Im(c^{k-m})\sin\left(\tfrac{1}{2}\pi s% \right)\right){\Gamma^{(r)}}\left(s\right){\zeta^{(m-r)}}\left(s\right),.$

where

 25.4.6 $c=-\ln\left(2\pi\right)-\tfrac{1}{2}\pi\mathrm{i}.$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\ln\NVar{z}$: principal branch of logarithm function and $c$ Referenced by: §25.6(ii) Permalink: http://dlmf.nist.gov/25.4.E6 Encodings: TeX, pMML, png See also: Annotations for 25.4 and 25