Do a simple test, unplug the start relay and without turning the key On measure the relay socket for 12 Volts at Relay (2) terminal 3 or Relay (3) terminal 5, either with a multimeter or a simple 12 Volt lamp.
If its had the Startus Interuptus fix it will be alive otherwise if it's only alive with the key On, that's bad.
[You must be registered and logged in to see this link.]BTW, this schematic along with all the other Guzzi schematics (all models) show the starter solenoid wrong, there are actually two coils inside every Guzzi solenoid not one. The solenoid will draw 50 Amps for a split second but it won't be able to do that if it has to go through the ignition switch, even with perfectly clean contacts the wiring will choke it down to ~ 30 Amps.
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Just to show you how the starter is supposed to crank over take a length of wire and crimp a spade connector on one end. Replace the wire to the spade connector on the starter solenoid. Making sure the bike is in Neutral, back wheel off the ground or with the clutch pulled in for safety touch the other end of the wire onto the battery positive terminal, if it turns over you have proven that there is nothing wrong with the starter, the battery and the battery connections.
Note you don't need the key turned on for this test.
If it doesn't turn over then you should test the battery connections, with the negative terminal of your meter fastened to a nice clean bolt.
While trying to crank over using this jumper lead measure the following
The battery Positive terminal - should be at least 10 Volts
The battery Negative terminal - should be close to zero while trying to crank, it will go negative if the ground is loose.
Actually before even measuring the connections pull all 3 off and give them a scrape and apply some Vaseline or even grease, in particular where the main negative bolts to the gearbox behind the starter.
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Why do the later Grisos have two Start relays, the early ones only had one and always crank, their single relay is always alive bypassing the ignition switch, my theory is the factory don't know it has two coils, not one schematic shows it and therefore don't know how much current it draws,
in fact they only fit a 15 Amp fuse, that also tells me they think it only draws a little current the second relay is a bodge to get around the weak yellow wire.
Note: if your wiring has been fixed with an always live relay you should upgrade to a 20 Amp fuse, the solenoid will now draw 50 Amps just for a few milliseconds like it was designed to do. A 20 amp is adequate for the very short time both coils are engaged.