Low-energy spin excitations have been studied on polycrystalline LaFeAsO1-xFx samples by inelastic neutron scattering. The Q-integrated dynamical spin susceptibility chi ''(omega) of the superconducting samples is found to be comparable to that of the magnetically ordered parent sample. On the other hand, chi ''(omega) almost vanishes at x = 0.158, where the superconducting transition temperature T-c is suppressed to 7 K. In addition, chi ''(omega) in optimally doped LaFeAsO0.918F0.082 with T-c = 29 K exhibits a spin resonance mode. The peak energy, E-res, when scaled by k(B)T(c) is similar to the value of about 4.7 reported in other high-T-c iron-based superconductors. This result suggests that there is intimate relationship between the dynamical spin susceptibility and high-T-c superconductivity in iron-based superconductors, and is consistent with a nesting condition between Fermi surfaces at the Gamma and M points. (C) 2011 Elsevier B.V. All rights reserved.