You'll find a fuel injection manual for the Gamma at
http://lanciagammaconsortium.info/smf1021/index.php/topic,156.0.htmlBut what you say doesn't seem right to me! The Gamma fuel injection system does not prime when the ignition is turned on; or at least it doesn't normally. If you could hear the pump running as soon as you turn the ignition on, theen either there's a fault or someone has changed things to overcome a problem somewhere else.
Power to the fuel pump in the Gamma is controlled by the double-relay. The fuse to the fuel pump is not in the fuse box but an in-line fuse hidden underneath the radiator expansion tank. The fuse connections often fail due to spilled water either expelled from the header tank when the engine gets too hot or by spilling it when refilling the system. So first thing to check is this fuse and its connections.
The double-relay has two functions, hence why it's double. First part is energised by a feed from the starter solenoid so that the fuel pump runs while cranking the engine. Once the engine has started and there's no power from the starter, power to the second part of the relay is via a micro-switch in the air metering box. When the engine is running the flap in the metering unit is opened by the air being sucked in by the engine and a switch connected to the air metering flap keeps the fuel pump running.
So first thing to check is the fuse under the expansion tank. If that's ok, then remove the bellows pipe from the air metering unit to the air filter housing so you can reach the air flap in the box. With the ignition on, move the flap with your finger or suitable stick and you should hear the whoosh of the fuel in the pipes and the pump going. If you still don't hear anything, then I'd suspect the pump. Put a voltmeter across the pump to see if it has power.