§25.4 Reflection Formulas

For $s\ne 0,1$,

25.4.1		$$\zeta \left(1-s\right)=2{(2\pi )}^{-s}\cos \left(\frac{1}{2}\pi s\right)\mathrm{\Gamma }\left(s\right)\zeta \left(s\right),$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function and $s$: complex variable Keywords: reflection formula Source: Apostol (1976, (12), p. 259) Referenced by: §25.10(i), §5.17 Permalink: http://dlmf.nist.gov/25.4.E1 Encodings: TeX, pMML, png See also: Annotations for §25.4 and Ch.25

25.4.2		$$\zeta \left(s\right)=2{(2\pi )}^{s-1}\sin \left(\frac{1}{2}\pi s\right)\mathrm{\Gamma }\left(1-s\right)\zeta \left(1-s\right).$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function and $s$: complex variable Keywords: reflection formula Source: Apostol (1976, (12), p. 259) A&S Ref: 23.2.6 Permalink: http://dlmf.nist.gov/25.4.E2 Encodings: TeX, pMML, png See also: Annotations for §25.4 and Ch.25

Equivalently,

25.4.3		$$\xi \left(s\right)=\xi \left(1-s\right),$$
ⓘ Symbols: $\xi \left(s\right)$: Riemann’s $\xi$-function and $s$: complex variable Keywords: reflection formula Source: Apostol (1976, p. 260) Permalink: http://dlmf.nist.gov/25.4.E3 Encodings: TeX, pMML, png See also: Annotations for §25.4 and Ch.25

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

25.4.4		$$\xi \left(s\right)=\frac{1}{2}s(s-1)\mathrm{\Gamma }\left(\frac{1}{2}s\right){\pi }^{-s/2}\zeta \left(s\right).$$
ⓘ Defines: $\xi \left(s\right)$: Riemann’s $\xi$-function Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter and $s$: complex variable Keywords: definition Source: Apostol (1976, p. 260) Permalink: http://dlmf.nist.gov/25.4.E4 Encodings: TeX, pMML, png See also: Annotations for §25.4 and Ch.25

For $s\ne 0,1$ and $k=1,2,3,\mathrm{\dots }$,

25.4.5		$${(-1)}^k{\zeta }^{(k)}\left(1-s\right)=\frac{2}{{(2\pi )}^s}\sum _{m=0}^k\sum _{r=0}^m\begin{array}{l}\left(\genfrac{}{}{0.0pt}{}{k}{m}\right)\left(\genfrac{}{}{0.0pt}{}{m}{r}\right)\left(\mathrm{\Re }\left(c^{k-m}\right)\cos \left(\frac{1}{2}\pi s\right)+\mathrm{\Im }\left(c^{k-m}\right)\sin \left(\frac{1}{2}\pi s\right)\right)\\ \times {\mathrm{\Gamma }}^{(r)}\left(s\right){\zeta }^{(m-r)}\left(s\right),\end{array}$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{\Im }$: imaginary part, $\mathrm{\Re }$: real part, $\sin z$: sine function, $k$: nonnegative integer, $m$: nonnegative integer, $s$: complex variable and $c$ Keywords: derivative Source: Apostol (1985a, (6), p. 224) Notes: See Apostol (1985a) Referenced by: §25.4 Permalink: http://dlmf.nist.gov/25.4.E5 Encodings: TeX, pMML, png See also: Annotations for §25.4 and Ch.25

where

25.4.6		$$c\equiv -\ln \left(2\pi \right)-\frac{1}{2}\pi \mathrm{i}.$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\equiv$: equals by definition, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function and $c$ Keywords: definition Source: Apostol (1985a, p. 223) Referenced by: §25.6(ii) Permalink: http://dlmf.nist.gov/25.4.E6 Encodings: TeX, pMML, png See also: Annotations for §25.4 and Ch.25