If you have a multimeter that can measure current you could try putting the battery in series with the multimeter and measure the current draw of the L-328. First, the current draw should go to zero (or very close to it) when the power is shut off. Second, the current draw should be reasonable. Third, if it has a battery test feature and the current draw increases during the test, it should drop back down after the test is complete.
Does the battery quickly die even with the power off, or only when you leave it on for a few solid hours?
Denis K