^L usually has no assigned action.

  $ stty -a | grep -F '^D'
  intr = ^C; quit = ^\; erase = ^?; kill = ^U; eof = ^D; eol = <undef>;

  $ stty -a | grep -F '^L'
  $

Your Ctrl-L clear-and-refresh command is implemented in Bash (or rather GNU readline).

There is a "reprint" action, commonly bound to ^R:

  $ stty -a | grep rprnt
  eol2 = <undef>; swtch = <undef>; start = ^Q; stop = ^S; susp = ^Z; rprnt = ^R;

It doesn't clear the screen; it just issues a newline and reprints the characters buffered so far. You can use stty to assign ^L to "rprnt".