For those who are having trouble starting the game on Windows 10:
Completely close out Steam. Kill it in task manager.
Run Steam as Admin.
Start L.A. Noire.

What's happening is the Steam package for the game isn't being run as admin when you click play, and it isn't able to properly install. Running Steam as Admin fixes this.

This fixed the problem I was having where clicking Launch Game from launcher resulted in the window closing and nothing happening with no errors.

For those who are having trouble starting the game on Windows 10: Completely close out Steam. Kill it in task manager. Run Steam as Admin. Start L.A. Noire. What's happening is the Steam package for the game isn't being run as admin when you click play, and it isn't able to properly install. Running Steam as Admin fixes this. This fixed the problem I was having where clicking Launch Game from launcher resulted in the window closing and nothing happening with no errors.