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There are several temperature gauges in the LT1: digital oil, analog oil, digital water, and analog water. Which one is not working on yours? The digital oil gauge will throw a code if it isn't working, 30 minutes after start-up. You can temporarily clear the code by shutting off the engine and restarting it.
I'm going to assume you are talking about the analog water temperature gauge since it is always looking at you and active. The sensor for it I believe is on the right side of the engine, in the head between the last 2 cylinders. See if the wire is connected. It's unlikely the gauge in the dash is bad.
There is also a digital read out coolant temperature reading in the center of the speedometer. You can toggle thru the different readings of coolant temperature, oil temperature and charging volts by pressing the GAUGES button and releasing it. Stop at the coolant temperature. That is what you should go by and what the engine sees for ECM input for engine operation. That is a separate sensor and it is in the front of the water pump.
I'm going to assume you are talking about the analog water temperature gauge . The sensor for it I believe is on the right side of the engine, in the head between the last 2 cylinders. See if the wire is connected. .
Remove the wire and ground it on the head.If temp gauge moves then most likely temp sending unit is bad