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