Let's rest for a moment - perhaps I'm off the base.
Do both groupheads react the same? Are you simply using the manual brew button or are you trying all of the dose buttons?
That particular relay is, perhaps, in the espresso world, unique to LM and you really should have ordered the part from an LM distributor - here is a good source - http://www.espressoparts.com/
. To the best of my knowledge, WLL does not deal in LM machines or their parts. That specific relay may or may not be available as a replacement part - if LM owners were lucky, that relay would simply plug into a socket but that's wishful thinking.
Secondly, ditto what KDM said.
edit - Thirdly - here is the relay - if, in fact, that is the problem - http://www.marzoccoman.com/servlet/the-93/Pump-Relay-%28Relay%29/Detail