This could be any of several common causes of overheating, of any water cooled engine. I have not owned my LS long enough to have a heat problem so based on experience with other cars and reading this website...
Things to suspect are Thermostat, Clogged cooling passages, malfunctioning fan, clogged air fins on radiator, bad sensors, clogged exhaust system, running too lean(engine control issue), dragging drivetrain components and on and on.
So, tell us more. What else does your car do? Run rough, any other check engine signals? Smoke, steam?
Most likely causes in my opinion would be Thermostat, especially if it has never been changed. Fixing the T stat "should" be cheap but with our "unique" engines, the part may be more expensive than normal.
From reading this webpage, the early LS had an oil pump driven fan, what Rube Goldberg did that? Anyway, the fan apparantly has been prone to failure, so it is worth checking. I have read that fixing this fan is not cheap.
The things that are free and easy to check are the dirty fins on the radiator, whether the fan is running and when and how fast it is running, and maybe collapsed radiator hoses.
Good Luck, and tell us any other symptoms.
Jim Henderson