We present an inelastic neutron scattering study of the spin dynamics in the mixed phase of the high-T-c superconductor La2-xSrxCuO4, In the overdoped regime (x = 0. 17, T-c = 37 K) the application of a magnetic field induces a spectral weight redistribution centered around the zero-field spin gap (Delta(SG) approximate to 5 6.5 meV). Moreover the opening of the spin gap seems to track the irreversibility temperature (related to the melting of the vortex lattice) rather than T-c, indicating an unusual interplay between the mesoscopic and microscopic properties in La2-xSrxCuO4. In the underdoped regime (x = 0. 10, T-c = 29 K) our zero-field data indicate a strong decrease (or even vanishing) of the spin gap in the superconducting state and no dependence upon the application of a magnetic field.