The Zeta function (zeta function) is defined for Re(z)>1 by

and is extended to the rest of the complex plane (except for the point z=1) by analytic continuation.  The point z=1 is a simple pole.

The call Zeta(n, z) gives the nth derivative of the Zeta function,

You can enter the command Zeta using either the 1-D calling sequence or in 2-D using command completion.

The optional third parameter v changes the expression of summation to 1/(i+v)^z, so that for Re(z)>1,

and, again, this is extended to the complex plane less the point 1 by analytic continuation.  The point z=1 is a simple pole for the function Zeta(0, z, v).

The third parameter, v, can be any complex number which is not a non-positive integer.

The function Zeta(0, z, v) is often called the Hurwitz Zeta function or the Generalized Zeta function.

$\mathrm{\zeta }\left(2.2\right)$

$1.490543257$

$\mathrm{evalf}\left(\mathrm{\zeta }\left(-1.5+3.5\mathrm{I}\right)\,30\right)$

$0.232434139233841813873124398558+0.173728378830616590886617515292\mathrm{I}$

${\mathrm{\zeta }}^{\left(1\right)}\left(\frac{1}{2}\right)$

$\mathrm{\zeta }\left(\frac{1}{2}\right)\left(\frac{\mathrm{\gamma }}{2}+\frac{\ln \left(8\mathrm{\pi }\right)}{2}+\frac{\mathrm{\pi }}{4}\right)$

${\mathrm{\zeta }}^{\left(0\right)}\left(2\,\frac{1}{2}\right)$

$\frac{{\mathrm{\pi }}^2}{2}$

${\mathrm{\zeta }}^{\left(0\right)}\left(2\,s\right)$

$\mathrm{\Psi }\left(1\,s\right)$

${\mathrm{\zeta }}^{\left(3\right)}\left(1.5+0.3\mathrm{I}\,0.2\right)$

$70.20062910+64.74329586\mathrm{I}$

${\mathrm{\zeta }}^{\left(3\right)}\left(-1.2+35.3\mathrm{I}\,0.2+\mathrm{I}\right)$

$\mathrm{-2.383200150}\times {10}^{21}+1.841204211\times {10}^{21}\mathrm{I}$

$\mathrm{sum}\left(\frac{1}{i^7}\,i=1..\mathrm{\infty }\right)$

$\mathrm{\zeta }\left(7\right)$

$\mathrm{plots}:-\mathrm{complexplot}\left(\mathrm{\zeta }\left(0.5+t\mathrm{I}\right)\,t=0..34\,\mathrm{scaling}=\mathrm{constrained}\,\mathrm{numpoints}=300\,\mathrm{labels}=\left[''Re''\,''Im''\right]\right)$

Erdelyi, A. Higher Transcendental Functions. McGraw-Hill, 1953. Vol. 1.