Since its launch in 2008, the Fermi Gamma-ray Burst Monitor (GBM) has triggered and located on average approximately two gamma-ray bursts (GRB) every three days. Here we present the third of a series of catalogs of GRBs detected by GBM, extending the second catalog by two more years, through the middle of July 2014. The resulting list includes 1405 triggers identified as GRBs. The intention of the GBM GRB catalog is to provide information to the community on the most important observables of the GBM detected GRBs. For each GRB the location and main characteristics of the prompt emission, the duration, peak flux and fluence are derived. The latter two quantities are calculated for the 50-300~keV energy band, where the maximum energy release of GRBs in the instrument reference system is observed, and also for a broader energy band from 10-1000 keV, exploiting the full energy range of GBMs low-energy NaI(Tl) detectors. Using statistical methods to assess clustering, we find that the hardness and duration of GRBs are better fitted by a two-component model with short-hard and long-soft bursts, than by a model with three components. Furthermore, information is provided on the settings and modifications of the triggering criteria and exceptional operational conditions during years five and six in the mission. This third catalog is an official product of the Fermi GBM science team, and the data files containing the complete results are available from the High-Energy Astrophysics Science Archive Research Center (HEASARC).
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