Recently, a tight correlation among three quantities that characterize the prompt emission of long Gamma-Ray Bursts (GRBs) with known redshift z, was discovered (Firmani et al. 2006). We use this correlation to construct the Hubble diagram (HD) with a sample of 19 GRBs in the broad range of z=0.17-4.5, and carry out a full statistical analysis to constrain cosmological parameters (CPs). To optimally solve the problem of circularity, a Bayesian approach is applied. The main result is that the concordance LambdaCDM cosmology is fully consistent with the GRB data at the level of several tests. If we assume the Lambda cosmology, then we find Om_M=0.31^{+0.09}_{-0.08} and Om_Lambda=0.80^{+0.20}_{-0.30}$ (1sigma); the flat-geometry case is within 1sigma. Assuming flatness, we find Om_M=0.29^{+0.08}_{-0.06}, and fixing Om_M=0.28, we obtain a dark energy equation of state parameter w=-1.07^{+0.25}_{-0.38}, i.e. the ambdaCDM model (w=-1) is within 1sigma. Given the low number of usable GRBs we cannot yet constrain well the possible evolution of w=w(z). However, the case w(z)=-1 (LambdaCDM) is consistent at the 68.3% CL with GRBs. It is shown also how a broad range of zs in the used sample improves the determination of CPs from the HD, which is the case of GRBs as distance indicators.
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