My Battery light comes on intermmediately

My battery light on 2000 Discovery recently has been coming when I start up engine. However, it usually turns off after I rev up the engine to about 3500 rpm's. The times when it the light doesn't turn off, it usually turns off after I get on the freeway.

Does this mean my alternator is going bad, or does my battery need to be replaced? Or is there something else I need to look at or replace?

Any info on this matter would be helpful.

So I took it to advance auto for a charging system check: Battery came back at 12.77 with 634 cca; Starter came back at voltage 10.57v, Amps 0.0a, 2.67seconds; Alternator no load 13.910 and with load 13.86. walk into store come back out to start and the battery light is pressed the accelerator and it goes away, I don't want to spend the money if it is the wiring how do I figure it out?

Please help me!!!!