Here is the full list of all 13 LA Cops achievements worth 1,000 gamerscore. It takes around 6-8 hours to unlock all of the achievements on Xbox One.
Finish any level without losing any Cops.
Arrest all criminals in any level.
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.
Complete all the game missions in Nightmare Mode.
Complete all the game missions in Hardcore Mode.
Fully upgrade any Cop.