# §25.6 Integer Arguments

## §25.6(i) Function Values

 25.6.1 $\displaystyle\mathop{\zeta\/}\nolimits\!\left(0\right)$ $\displaystyle=-\frac{1}{2},$ $\displaystyle\mathop{\zeta\/}\nolimits\!\left(2\right)$ $\displaystyle=\frac{\pi^{2}}{6},$ $\displaystyle\mathop{\zeta\/}\nolimits\!\left(4\right)$ $\displaystyle=\frac{\pi^{4}}{90},$ $\displaystyle\mathop{\zeta\/}\nolimits\!\left(6\right)$ $\displaystyle=\frac{\pi^{6}}{945}.$ Symbols: $\mathop{\zeta\/}\nolimits\!\left(\NVar{s}\right)$: Riemann zeta function and $\pi$: the ratio of the circumference of a circle to its diameter A&S Ref: 23.2.11 Referenced by: §25.6(i) Permalink: http://dlmf.nist.gov/25.6.E1 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for 25.6(i)
 25.6.2 $\displaystyle\mathop{\zeta\/}\nolimits\!\left(2n\right)$ $\displaystyle=\frac{(2\pi)^{2n}}{2(2n)!}\left|B_{2n}\right|,$ $n=1,2,3,\dots$. 25.6.3 $\displaystyle\mathop{\zeta\/}\nolimits\!\left(-n\right)$ $\displaystyle=-\frac{B_{n+1}}{n+1},$ $n=1,2,3,\dots$. Symbols: $B_{\NVar{n}}$: Bernoulli numbers, $\mathop{\zeta\/}\nolimits\!\left(\NVar{s}\right)$: Riemann zeta function and $n$: nonnegative integer A&S Ref: 23.2.15 (in slightly different form) Permalink: http://dlmf.nist.gov/25.6.E3 Encodings: TeX, pMML, png See also: Annotations for 25.6(i) 25.6.4 $\displaystyle\mathop{\zeta\/}\nolimits\!\left(-2n\right)$ $\displaystyle=0,$ $n=1,2,3,\dots$. Symbols: $\mathop{\zeta\/}\nolimits\!\left(\NVar{s}\right)$: Riemann zeta function and $n$: nonnegative integer A&S Ref: 23.2.14 Referenced by: §25.6(i) Permalink: http://dlmf.nist.gov/25.6.E4 Encodings: TeX, pMML, png See also: Annotations for 25.6(i)
 25.6.5 $\mathop{\zeta\/}\nolimits\!\left(k+1\right)=\frac{1}{k!}\sum_{n_{1}=1}^{\infty% }\dots\sum_{n_{k}=1}^{\infty}\frac{1}{n_{1}\cdots n_{k}(n_{1}+\dots+n_{k})},$ $k=1,2,3,\dots$.
 25.6.6 $\mathop{\zeta\/}\nolimits\!\left(2k+1\right)=\frac{(-1)^{k+1}(2\pi)^{2k+1}}{2(% 2k+1)!}\int_{0}^{1}\mathop{B_{2k+1}\/}\nolimits\!\left(t\right)\mathop{\cot\/}% \nolimits\!\left(\pi t\right)\mathrm{d}t,$ $k=1,2,3,\dots$.
 25.6.7 $\displaystyle\mathop{\zeta\/}\nolimits\!\left(2\right)$ $\displaystyle=\int_{0}^{1}\int_{0}^{1}\frac{1}{1-xy}\mathrm{d}x\mathrm{d}y.$ 25.6.8 $\displaystyle\mathop{\zeta\/}\nolimits\!\left(2\right)$ $\displaystyle=3\sum_{k=1}^{\infty}\frac{1}{k^{2}\binom{2k}{k}}.$ Symbols: $\mathop{\zeta\/}\nolimits\!\left(\NVar{s}\right)$: Riemann zeta function, $\binom{\NVar{m}}{\NVar{n}}$: binomial coefficient and $k$: nonnegative integer Referenced by: §25.6(i) Permalink: http://dlmf.nist.gov/25.6.E8 Encodings: TeX, pMML, png See also: Annotations for 25.6(i) 25.6.9 $\displaystyle\mathop{\zeta\/}\nolimits\!\left(3\right)$ $\displaystyle=\frac{5}{2}\sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{k^{3}\binom{2k}{% k}}.$ Symbols: $\mathop{\zeta\/}\nolimits\!\left(\NVar{s}\right)$: Riemann zeta function, $\binom{\NVar{m}}{\NVar{n}}$: binomial coefficient and $k$: nonnegative integer Permalink: http://dlmf.nist.gov/25.6.E9 Encodings: TeX, pMML, png See also: Annotations for 25.6(i) 25.6.10 $\displaystyle\mathop{\zeta\/}\nolimits\!\left(4\right)$ $\displaystyle=\frac{36}{17}\sum_{k=1}^{\infty}\frac{1}{k^{4}\binom{2k}{k}}.$ Symbols: $\mathop{\zeta\/}\nolimits\!\left(\NVar{s}\right)$: Riemann zeta function, $\binom{\NVar{m}}{\NVar{n}}$: binomial coefficient and $k$: nonnegative integer Referenced by: §25.6(i) Permalink: http://dlmf.nist.gov/25.6.E10 Encodings: TeX, pMML, png See also: Annotations for 25.6(i)

## §25.6(ii) Derivative Values

 25.6.11 $\mathop{\zeta\/}\nolimits'\!\left(0\right)=-\tfrac{1}{2}\mathop{\ln\/}% \nolimits\!\left(2\pi\right).$ Symbols: $\mathop{\zeta\/}\nolimits\!\left(\NVar{s}\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter and $\mathop{\ln\/}\nolimits\NVar{z}$: principal branch of logarithm function A&S Ref: 23.2.13 Permalink: http://dlmf.nist.gov/25.6.E11 Encodings: TeX, pMML, png See also: Annotations for 25.6(ii)
 25.6.12 $\mathop{\zeta\/}\nolimits''\!\left(0\right)=-\tfrac{1}{2}(\mathop{\ln\/}% \nolimits\!\left(2\pi\right))^{2}+\tfrac{1}{2}{\gamma^{2}}-\tfrac{1}{24}\pi^{2% }+\gamma_{1},$

where $\gamma_{1}$ is given by (25.2.5).

With $c$ defined by (25.4.6) and $n=1,2,3,\dots,$

 25.6.13 $\displaystyle(-1)^{k}{\mathop{\zeta\/}\nolimits^{(k)}}\!\left(-2n\right)$ $\displaystyle=\frac{2(-1)^{n}}{(2\pi)^{2n+1}}\sum_{m=0}^{k}\sum_{r=0}^{m}% \binom{k}{m}\binom{m}{r}\Im{(c^{k-m})}\*{\mathop{\Gamma\/}\nolimits^{(r)}}\!% \left(2n+1\right){\mathop{\zeta\/}\nolimits^{(m-r)}}\!\left(2n+1\right),$ 25.6.14 $\displaystyle(-1)^{k}{\mathop{\zeta\/}\nolimits^{(k)}}\!\left(1-2n\right)$ $\displaystyle=\frac{2(-1)^{n}}{(2\pi)^{2n}}\sum_{m=0}^{k}\sum_{r=0}^{m}\binom{% k}{m}\binom{m}{r}\Re{(c^{k-m})}\*{\mathop{\Gamma\/}\nolimits^{(r)}}\!\left(2n% \right){\mathop{\zeta\/}\nolimits^{(m-r)}}\!\left(2n\right),$ 25.6.15 $\displaystyle\mathop{\zeta\/}\nolimits'\!\left(2n\right)$ $\displaystyle=\frac{(-1)^{n+1}(2\pi)^{2n}}{2(2n)!}\left(2n\mathop{\zeta\/}% \nolimits'\!\left(1-2n\right)-(\mathop{\psi\/}\nolimits\!\left(2n\right)-% \mathop{\ln\/}\nolimits\!\left(2\pi\right))B_{2n}\right).$

## §25.6(iii) Recursion Formulas

 25.6.16 $\left(n+\tfrac{1}{2}\right)\mathop{\zeta\/}\nolimits\!\left(2n\right)=\sum_{k=% 1}^{n-1}\mathop{\zeta\/}\nolimits\!\left(2k\right)\mathop{\zeta\/}\nolimits\!% \left(2n-2k\right),$ $n\geq 2$. Symbols: $\mathop{\zeta\/}\nolimits\!\left(\NVar{s}\right)$: Riemann zeta function, $k$: nonnegative integer and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/25.6.E16 Encodings: TeX, pMML, png See also: Annotations for 25.6(iii)
 25.6.17 $\left(n+\tfrac{3}{4}\right)\mathop{\zeta\/}\nolimits\!\left(4n+2\right)=\sum_{% k=1}^{n}\mathop{\zeta\/}\nolimits\!\left(2k\right)\mathop{\zeta\/}\nolimits\!% \left(4n+2-2k\right),$ $n\geq 1$. Symbols: $\mathop{\zeta\/}\nolimits\!\left(\NVar{s}\right)$: Riemann zeta function, $k$: nonnegative integer and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/25.6.E17 Encodings: TeX, pMML, png See also: Annotations for 25.6(iii)
 25.6.18 ${\left(n+\tfrac{1}{4}\right)\mathop{\zeta\/}\nolimits\!\left(4n\right)+\tfrac{% 1}{2}(\mathop{\zeta\/}\nolimits\!\left(2n\right))^{2}=\sum_{k=1}^{n}\mathop{% \zeta\/}\nolimits\!\left(2k\right)\mathop{\zeta\/}\nolimits\!\left(4n-2k\right% )},$ $n\geq 1$. Symbols: $\mathop{\zeta\/}\nolimits\!\left(\NVar{s}\right)$: Riemann zeta function, $k$: nonnegative integer and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/25.6.E18 Encodings: TeX, pMML, png See also: Annotations for 25.6(iii)
 25.6.19 $\left(m+n+\tfrac{3}{2}\right)\mathop{\zeta\/}\nolimits\!\left(2m+2n+2\right)=% \left(\sum_{k=1}^{m}+\sum_{k=1}^{n}\right)\mathop{\zeta\/}\nolimits\!\left(2k% \right)\mathop{\zeta\/}\nolimits\!\left(2m+2n+2-2k\right),$ $m\geq 0$, $n\geq 0$, $m+n\geq 1$.
 25.6.20 $\tfrac{1}{2}(2^{2n}-1)\mathop{\zeta\/}\nolimits\!\left(2n\right)=\sum_{k=1}^{n% -1}(2^{2n-2k}-1)\mathop{\zeta\/}\nolimits\!\left(2n-2k\right)\mathop{\zeta\/}% \nolimits\!\left(2k\right),$ $n\geq 2$. Symbols: $\mathop{\zeta\/}\nolimits\!\left(\NVar{s}\right)$: Riemann zeta function, $k$: nonnegative integer and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/25.6.E20 Encodings: TeX, pMML, png See also: Annotations for 25.6(iii)

For related results see Basu and Apostol (2000).