  Effective spin foams provide the computationally most efficient spin foam
models yet and are therefore ideally suited for applications e.g. to quantum
cosmology. We provide here the first effective spin foam computations of a
finite time evolution step in a Lorentzian quantum de Sitter universe. We will
consider a set-up which computes the no-boundary wave function, as well as a
set-up describing the transition between two finite scale factors. A key
property of spin foams is that they implement discrete spectra for the areas.
We therefore study the effects that are induced by the discrete spectra.
  To perform these computations we had to identify a technique to deal with
highly oscillating and slowly converging, or even diverging sums. We illustrate
here that high order Shanks transformation work very well and are a promising
tool for the evaluation of Lorentzian (gravitational) path integrals and spin
foam sums.
