Dynamic structure factors for a lipid bilayer have been calculated from molecular dynamics simulations. From trajectories of a system containing 1024 lipids we obtain wave vectors down to 0.34 nm(-1), which enables us to directly resolve the Rayleigh and Brillouin lines of the spectrum. The results confirm the validity of a model based on generalized hydrodynamics, but also improves the line widths and the position of the Brillouin lines. The improved resolution shows that the Rayleigh line is narrower than in earlier studies, which corresponds to a smaller thermal diffusivity. From a detailed analysis of the power spectrum, we can, in fact, distinguish two dispersive contributions to the elastic scattering. These translate to two exponential relaxation processes in separate time domains. Further, by including a first correction to the wave-vector-dependent position of the Brillouin lines, the results agree favorably to generalized hydrodynamics even up to intermediate wave vectors, and also yields a 20% higher adiabatic sound velocity. The width of the Brillouin lines shows a linear, not quadratic, dependence to low wave vectors.