Is it possible to name the 'screen' logfile from the -L flag?
Probably the easiest way is to use the following:
screen -L -Logfile log_filename your_command
You can edit your .screenrc file and add something like this:
logfile /tmp/myownlog
It is also answered at Specifying a log name for screen output without relying on .screenrc.
Alternatively, there's a way to do it online.
Enter command mode in Screen via Ctrl+a, : and use the logfile command with the name of the file you want as argument, for example:
logfile whatevernameyoulike.logSource: Screen man page