I will chime in also. Lufthansa to Lufthansa is not bad in Frankfurt. Lufthansa uses only A and B terminals, You will come in to Terminal A, and you will likely leave from A as this is also a Schengen flight. With luck, you will not gate but be bused from the tarmac, as the bus drops you practically at immigration control. Keep in mind that if your plane leaves LAX timely (I know nothing about how well or poorly LAX functions), you will have a good chance to land as much as 30 minutes land ahead of schedule. We have taken 6 round trips through Frankfurt from Philadelphia with ongoing flights since 2014, more often than not we have been bused in, we have never had an issue with our connections, even making one that had been reduced from 90 minutes to 15 because of problems with our plane delayed first coming to Philadelphia. And on that flight, Lufthansa had already re-booked numerous passengers before we landed. Actually, the only one we ever sweated was our return this spring from Venice, that had been rescheduled by Lufthansa from 80 to only 70 minutes, was late out of Venice, and there was a gigantic line at the Schengen passport exit. We barely made that one, luckily that plane was a bit delayed and loaded late.
Relax and go for it. I will bet you will even have enough ti me to grab a sandwich or pastry on the way to your gate.