The only definition that I have ever seen of this Riemann-Hurtwitz zeta-function is this,

For $0 < a \leq 1$ we have the identity $$ \zeta(z, a) = \frac{2 \Gamma(1 - z)}{(2 \pi)^{1-z}} \left[\sin \frac{z \pi}{2} \sum_{n=1}^\infty \frac{\cos 2 \pi a n}{n^{1-z}} + \cos \frac{z \pi}{2} \sum_{n=1}^\infty \frac{\sin 2 \pi a n}{n^{1-z}} \right] \,$$

Now from this (or otherwise!) how do I show that, $\xi(s,\frac{1}{2}) = \sum_{n=0}^{\infty} \frac{1}{ (n + \frac{1}{2} )^s }$

It seems that for any $k$ even and $d$ a positive integer there is such a sum over inverse powers representation of $\xi(s-k,\frac{d}{2}-1 )$ - it would be great to know if there is a general expression here!

Any insight/motivation about this $\xi(z,a)$ function would be a great help.