The Lyapunov vectors corresponding to the steps of Lyapunov spectra for many-particle systems show time-oscillating behavior in two types of Lyapunov modes, one associated with time-translational invariance and the other with spatial translational invariance. Our result is that, for each coordinate direction, the longest period of the Lyapunov modes is twice as long as the period of the momentum autocorrelation function. A simple explanation for this relation is proposed and we argue that this result is generally true for many-particle systems. This gives the first quantitative connection between the Lyapunov modes and an experimentally accessible quantity.