Did you try repair the game? (Its button is located in the launcher)
EDIT : Found this on *PBE forums
*PBE: I changed the directories.
Not sure if it works.
1. Go into your League of Legends directory
2. Go to the RADS folder, then into the projects folder
3. Delete lol_launcher, lol_air_client, and then lol_air_client_config_na
4. Open the League of Legends launcher and wait for it to reinstall. (takes a while)