Here is the solenoid and where it is located.
If you suspect the solenoid, (which is an unlikely failure), remove it briefly. Remember then you cannot stop the engine with the key!
I think you will be more productive on checking out the fuel filter for water and ice.
When did you buy the diesel fuel you are currently using?
The only free WSM for an L3400 is from Scotland and may not represent the safety controls your tractor has concerning the stop solenoid.
From looking at other tractor models the same part fits, I believe the stop solenoid on your engine has two internal coils. a Pull in coil to actually move the plunger and a hold coil to keep it retracted once moved by the pull in coil.
Before removing the actual solenoid I suggest you check for power at the solenoid electrical terminal. Unplug the connector from the solenoid and check for power on the wiring harness connecor using a test light in both the ON and the START positions of the key switch.
Dave