How can I check whether my phone is network locked (Samsung GALAXY S3)? I have a Samsung GALAXY S3 with a rooted Android 4.1.2. My network provider is AT&amp;T.

I tried dialing *#7465625# but it doesn't work:
You could try an app to do this, Samsung have removed/changed some of the Dialer codes on the Galaxy range. 

GalaxSim Unlock seems to do what you want, it checks if SIM locked, and offers an in-app purchase to SIM unlock. Please note I have not tested this app. 

You could also pop in a SIM from a friend's device that is on another network and that should tell you pretty quick!