An L-glutamate ligase that catalyzes the ATP-dependent post-translational addition of glutamate residues to the C-terminus of ribosomal protein bS6 (RpsF). Is also able to catalyze the synthesis of poly-alpha-glutamate in vitro, via ATP hydrolysis from unprotected glutamate as substrate. The number of glutamate residues added to either RpsF or to poly-alpha-glutamate changes with pH