Usually the charging system ( the battery ) and the ignition system ( spark plugs ) are totally independent.
However on some of the more modern systems there is circuitry that controls the spark timing and this needs battery power, not much but some power just the same.
Also there is a fuel cut off solenoid under the carb and most new ones need 10V or more to open & allow fuel to flow into the engine, so a dead flat battery will oft not be able to be jumped in the mower.
The L series are current so if this is a new mower, take it back for a warrantee repair.
Do not start to pull anything apart nor touch anything that is not in the routine maintenance section of the owners manual or good chance the warrantee will be voided.
If this is an old ( out of warrantee ) mower then go to the deere web site and look at the manual on line or better still buy a hard copy.
The free repair manual is not fully down loadable, you have to do it a few pages at a time, to encourage you to buy a copy, but is well worth the money.
It explains how each section works, what goes wrong, where each bit is located on your mower and also where each bit is located on the wiring loom.
Better still it details in step by step how to test the system and the order to test it.
It also breaks down the loom into individual sections for testing
So you have a set of tests for sparking and another set fr cranking then another for charging then another for powered accessories ( where fitted )
Now I could type 50 pages here, paraphrasing the manual but it still would not be as useful as the real item.
I know "go read the book" is not what you were hoping for but in the case of JD there is no better advice I could give.
If ou get stuck or have troubles understanding what you have read, feel free to ask for help.
Not every one can read nor understand simple circuit diagrams and a lot of males are colour blind to varing degrees so this does not help either.