We studied proton dynamics of a hydrogen bonds of the crystalline l-ascorbic acid. Our approach was based on the Car-Parrinello molecular dynamics. The focal point of our study was simulation of the infrared spectra of l-ascorbic acid associated with the O-H stretching modes that are very sensitive to the strength of hydrogen bonding. In the l-ascorbic acid there are four kinds of hydrogen bonds. We calculated their spectra by using anharmonic approximation and the time course of the dipole moment function as obtained from the Car-Parrinello simulation. The quantization of the nuclear motion of the protons was made to perform detailed analysis of strength and properties of hydrogen bonds. We presented double minimum proton potentials with small value of barriers for medium-strong hydrogen bonds. We have also shown the difference character of medium-strong hydrogen bonds compared to weaker hydrogen bonds in the l-ascorbic acid.