Your LS1 should not have an overheating problem if the standard cooling system is working the way it should - The stock radiator is large and aluminium and cools very well. Without the a/c on both of mine rarely switch on the secondary fan unless in stop start traffic.
Get it checked out, the system should be flushed and coolant changed at least once a year imo.
While we are on it, the LS1's cooling system is self bleeding so that will not be a problem either unlike the V6.
I would suspect the common things, fans (switch the a/c on and off when engine cold - you will hear the fans go on and off), water pump, thermostat, radiator blocked, etc.