A normalisation of the ordinary Dirichlet Series in the lower half complex plane that has the equivalent normalised Riemann Zeta function as an detrended envelope function.

In the lower half complex plane Re(s)<1, a convergent normalisation of the ordinary dirichlet series is given, with a functional dependence on the real axis of 1/(1-Re(s)). Off the real axis, the normalised series has an upper (lower) detrended envelope function of the form ±| ζ(s)/N^(1−Re(s))|, for 10000 < N < ∞.