I found this paragraph on page 19 of the Los Angeles Animal Services Permit Book, which can be downloaded as a PDF here.
I think you might also be required to license your chickens, but I've had a hard time finding out if that's really the case. And, honestly, unless you live in a neighborhood with no real crime, no feral cats and no stray dogs, I seriously doubt you will ever be asked for your chicken license. I have satisfied myself with sticking to the distance rules, cleaning the coop so it doesn't stink and having only hens, no roosters. My neighbors know about my chickens, but only because I've told them. Three hens don't really cause much trouble.
The other question I got was where to get chickens in Los Angeles. There are actually quite a few places. Just call around to feed stores in your area between February and August, and you'll probably find someone selling chicks.
I have purchased chickens from three places in the LA area, although I'd probably only recommend Agoura Feed in Agoura Hills. As a general rule, I suggest you get your chickens as babies--preferably only a couple of days old--from a place that seems clean and that keeps the chicks away from older fowl. Ideally, you'd get them from a place that has only baby chicks. That just seems safest, disease-wise, to me. You can also order from My Pet Chicken, which ships small quantities of chicks and guarantees that they will arrive alive.
Try to get the chicks vaccinated for Marek's, if you can. It might save some heartache later. I'd also say a minimum of 3 chickens is ideal. You'll probably end up with extra eggs in the fridge, but just give them to friends. My feeling is that, if you have 3, if something happens and you lose one, you're not in a bind with one lonely chicken.
Hope that helps!