As for as I know, there are no shops left in Southern California that will touch one. European Cycle Specialties in Garden Grove was the last shop that would work on them and they are gone. Lance of Rickey Racer who used to be in Burbank is dead and I was told back in 2018 that Pro Italia in Glendale, the old mechanics are either dead or retired. The shops in the San Diego area are all gone. As for Northern California, I don't know. Scott Porter is in Denver but that's a long way from Los Angeles to get your bike worked on.
There are not enough Laverda's left on the road here to support a shop to work on them.
Now before I can give advice, you say that you are new to Laverda's? I am assuming you just bought the bike? Is it a nice running bike that needs minor maintenance or was it sitting in someones backyard for 40 years and is a plile of rust?
I do work on them but am retired and only when I have the time will I take the job. I am in Southern California.
So what work does the bike need? If by chance it's a bad crankshaft, in a couple months I will have all the tooling to fix them.