L might not have sent any notes about suspecting Light or anything, but the SPK now knows about the Death Note and how Kira kills thanks to him.
This was a HUGE step forward in catching Kira.
It literally took 24 episodes for L to get that information, and that information is crucial, so it will definitely help Mello and Near.
MaxVSTheGames
2022-02-10 03:21:38 +0000 UTC
I might be wrong, but I think there are 3 Death Notes in the world at the moment.
-Misa's, which is still buried in that forest (Light told Misa to tear out some pages and bury it back)
-Light's, which is now in the hands of the Japanese police, the one Mello wants them to hand over
-Rem's, the one Light found right after L died. Nobody besides him knows about this one, meaning that Light would still have a Death Note even though the police would hand over theirs to Mello
2022-02-10 00:17:45 +0000 UTC
Light deleted L note in Ep 26, the "recap" was actually all the notes he had, and the message he left at the end was he last words.
The timer on the computer was his personal computer, we saw it before they start moving between hotels. We can assume that him or Watari were going to that computer time to time to reset the timer, and if that didn't happens the message to the orphanage is sent with the message "L is dead".