Here is the full list of all 13 LA Cops achievements.
Fully upgrade any Cop.
Kill 600 criminals.
Kill 400 criminals.
Kill 200 criminals.
Kill 100 criminals.
Arrest all criminals in any level.
Finish a level without any of your Cops getting hit.
Finish any level without losing any Cops.
Complete all 8 main story missions.
Complete all the game missions in normal mode.
Kill 1000 criminals.
Complete all the game missions in Hardcore Mode.
Complete all the game missions in Nightmare Mode.