Spin excitations of overcloped La2-xSrxCuO4 (0.25 <= x <= 0.30) have been studied by neutron scattering utilizing triple-axis and time-of-flight spectrometers. Low-energy excitation spectra (omega <= 10 meV) for all of the superconducting samples show a maximum around omega = 6 meV with a maximum value of dynamic spin susceptibility chi ''(omega) that decreases linearly with T-c as doping increases. The magnetic scattering cross section finally disappears at x = 0.30, coincident with the disappearance of bulk superconductivity. Time-of-flight measurements for x = 0.25 and 0.30 show markedly suppressed magnetic scattering intensities at intermediate energies (20 <= omega <= 80 meV) in contrast to those of the stripe-ordered La1.875Ba0.125CuO4, which shows large spectral weight near the saddle point of the "hour-glass" shape dispersion. These facts imply a cooperative correlation between the dynamic incommensurate state and the superconductivity. (c) 2007 Elsevier B.V. All rights reserved.