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Please allow me throw in a few more words.
Reading the datasheet carefully you will find that the currents so far discussed are maximum currents - maximum in the sense of among a large group of samples. The typical current is 0.05uA, even for the non-'L' variants. The L variants seem to be just screened to meet lower maximum current specs.
These currents are listed Under conditions of VDD = 5V and Ta=85C
On the last page you can see how IDDS (standby current) goes up exponentially with temperature, from about 200nA at 85C down to just 4nA at room temperature (20C).
So you may actually have trouble to even measure these 4nA with hobbyist measurement Equipment.
With such current your 125mAh battery will last 3000 years ... theoretically. In reality the battery has it's own leakage current we don't know here.
So unless your ammeter can really measure something of uA's your safe to use even the non 'L' parts.
Ah, and another point touched before: the battery is only serving current to the SRAM and one more CMOS logic gate 14011, IC25. While the synth is not powerd, the other parts are separated from that supply through D15.
Cheers