If you have a reasonably accurate voltmeter, connect it to a convenient source (the cigar lighter socket would do if you have one), and observe the readings with engine off, engine running, various loads turned on (as CB suggests). 13.8 is a good number for "engine running, no load." If it's much lower than that, I'd suspect regulator and/or alternator.
If the LV warning device is based on a zener and a lamp, it's possible that the thing itself is faulty.
Certainly something to investigate.