basically, even though Rem wrote L's name into his notebook, he
could not kill L as he had written himself into the notebook
earlier, to die 23 days later (the maximum allowed, known as the
"23-day rule" ), and hence no one can now kill him using the Death
Note since only the first instance comes into effect.
L faked death after seeing Watari die, as his computer is connected
remotely to Watari's life monitor, and both L and Light saw that
the computer warned that W's heartbeat had stopped. What Light said
during this time was recorded down, and he admitted that he is the
first Kira.
He tried to kill his father after that using the Death Note, but he
didn't noe that it was switched with a fake notebook which
obviously didn't work.
