If you want to run a really fast marathon (whatever that might be for you), consider postponing to another time. If you are injured just over three weeks before the start, you risk making it much worse with the marathon.
If the goal is to run LA and get to the finishing line, pmaper yourself. Focus on recovery and if that means using exercise bikes or elyptical trainers to avoid stressing the calf muscle, so be it. If you are in this slot, you have already made the decision that finishing, not time, is key.
Most people who have run marathons for several years are able to explain why starting injured is not a good idea. Marathons are demanding if you run them for more than completion, and even if you only want to finish in whatever the time, they can still be hard. Every weekend somewhere in the US you will be able to find a marathon. If you take a pulled muscle and really push it, you may have the perfect excuse not to run for several months. I don't think it is worth the risk.
That said, if you are bound to run LA, rest until you get to the start, and then treat the event as a long training run. Don't be afraid to walk a bit. It may not be glorious, but it keeps you moving forward. A long run this weekend or next is merely a good way to ensure that you don't finish. Whatever marginal improvement to raceday cardio-vascular fitness is much more than offset by the risk of aggravating the pulled calf muscle.
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