  We consider the Generalized Gibbs Ensemble (GGE) in the context of global
quantum quenches in XXZ Heisenberg spin chains. Embedding the GGE into the
Quantum Transfer Matrix formalism we develop an iterative procedure to fix the
Lagrange-multipliers and to calculate predictions for the long-time limit of
short-range correlators. The main idea is to consider truncated GGE's with only
a finite number of charges and to investigate the convergence of the numerical
results as the truncation level is increased. As an example we consider a
quantum quench situation where the system is initially prepared in the N\'eel
state and then evolves with an XXZ Hamiltonian with anisotropy Delta>1. We
provide predictions for short range correlators and gather numerical evidence
that the iterative procedure indeed converges. The results show that the system
retains memory of the initial condition, and there are clear differences
between the numerical values of the correlators as calculated from the purely
thermal and the Generalized Gibbs ensembles.
