We have performed ab initio calculations for the adsorption of L-cysteine on Ag(111) using density functional theory. We have focused on two possible adsorbed species: the L-cysteine radical (center dot S-CH2-CH-NH2-COOH) adsorbed almost flat at a bridge site, slightly displaced toward an fcc location, and the zwitterionic radical Z-cysteine (center dot S-CH2-CH-NH3+-COO-) adsorbed at a bridge site, shifted to a hcp site forming a (4 x 4) unit cell (theta = 0.06) and a (root 3 x root 3) R 30 degrees unit cell (theta = 0.33), respectively. Special attention has been paid to the electronic structure of the system. The adsorbate-silver bond formation has been exhaustively investigated by analyzing the density of states projected onto the different atoms of the molecule, and by charge density difference calculations. A complicated interplay between sp and d states of silver in the formation of bonds between the adsorbates and the surface has been found. The role of the carboxyl group in the interaction with the surface has been also analyzed.