# §25.4 Reflection Formulas

For $s\neq 0,1$,

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

Equivalently,

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

where $\mathop{\xi\/}\nolimits\!\left(s\right)$ is Riemann’s $\mathop{\xi\/}\nolimits$-function, defined by:

 25.4.4 $\mathop{\xi\/}\nolimits\!\left(s\right)=\tfrac{1}{2}s(s-1)\mathop{\Gamma\/}% \nolimits\!\left(\tfrac{1}{2}s\right)\pi^{-s/2}\mathop{\zeta\/}\nolimits\!% \left(s\right).$ Defines: $\mathop{\xi\/}\nolimits\!\left(\NVar{s}\right)$: Riemann’s $\mathop{\xi\/}\nolimits$-function Symbols: $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: gamma function, $\mathop{\zeta\/}\nolimits\!\left(\NVar{s}\right)$: 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

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

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

where

 25.4.6 $c=-\mathop{\ln\/}\nolimits\!\left(2\pi\right)-\tfrac{1}{2}\pi\mathrm{i}.$ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{\ln\/}\nolimits\NVar{z}$: 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