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 or 2-D calling sequence.  For example, Zeta(1, 1/2) is equivalent to .

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.

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