  Aims: In this paper we study the main spectral and temporal properties of
gamma-ray bursts (GRBs) observed by Fermi/GBM. We investigate these key
properties of GRBs in the rest-frame of the progenitor and test for possible
intra-parameter correlations to better understand the intrinsic nature of these
events. Methods: Our sample comprises 32 GRBs with measured redshift that were
observed by GBM until August 2010. 28 of them belong to the long-duration
population and 4 events were classified as short/hard bursts. For all of these
events we derive, where possible, the intrinsic peak energy in the $\nu
F_{\nu}$ spectrum (\eprest), the duration in the rest-frame, defined as the
time in which 90% of the burst fluence was observed (\tninetyrest) and the
isotropic equivalent bolometric energy (\eiso). Results: The distribution of
\eprest has mean and median values of 1.1 MeV and 750 keV, respectively. A
log-normal fit to the sample of long bursts peaks at ~800 keV. No high-\ep
population is found but the distribution is biased against low \ep values. We
find the lowest possible \ep that GBM can recover to be ~ 15 keV. The
\tninetyrest distribution of long GRBs peaks at ~10 s. The distribution of
\eiso has mean and median values of $8.9\times 10^{52}$ erg and $8.2 \times
10^{52}$ erg, respectively. We confirm the tight correlation between \eprest
and \eiso (Amati relation) and the one between \eprest and the 1-s peak
luminosity ($L_p$) (Yonetoku relation). Additionally, we observe a parameter
reconstruction effect, i.e. the low-energy power law index $\alpha$ gets softer
when \ep is located at the lower end of the detector energy range. Moreover, we
do not find any significant cosmic evolution of neither \eprest nor
\tninetyrest.
