Smaller subcompact tractors (of several brands) have a problem where attempted starts result in clicking but no cranking. Most of those are lack of a full 12v to the solenoid with all sorts of remedial actions you can imagine.[There are several threads on this for Kubota and John Deere, etc.] I have that disease with my BX2200 which seems to happen only if I am in a hurry and need it to start immediately.
Does your L4400 have high hours or sat outside a lot? I have a B2150 (bought used with suspected poor care) that sometimes will not crank even with a known good battery, etc. No clicking, no cranking. That one always responds to rapping it soundly on the starter body with a socket extension rod driven by a gentle hammer. Then starts right up. There are several threads speculating about what that starter problem is.
Yours sounds a little more like the "low voltage to the solenoid" variety though maybe one of the safety interlocks acting up. Suggest using a voltmeter at the small wire to the solenoid (might have to nick the insulation to get on it.) When you turn the key to start if you get zero volts then you have an interlock problem in one of several places like the seat or the transmission lever or the PTO, etc. If you get voltage but not quite full battery voltage you have the "voltage drop to the solenoid" disease more common on the smaller machines.