I'm editing this question heavily for clarity:

I am looking for methods to compute $\zeta(1+it)$, or the (partially) completed Riemann zeta function $$\pi^{-s/2}\Gamma(s/2)\zeta(s)$$ along the line Re$(s)=1$. Certainly as I have left out the $s(s-1)$ factor, there is a pole at $s=1$. I would like to know the behaviour of the function above and below that point.

In particular, are there known mean value estimates, or even exact formulas for the argument of the functions above, preferably for small values of $t$?