L fucked up also by immediately saying that Kira sprung the trap, and now he knows where he is. L should have kept it secret that they had narrowed it down from a global investigation.
To be fair, that was mostly to goad Kira into trying to kill him to see how much information about a person Kira needed in order to kill them. He was able to confirm that Kira needed an identity in order to kill. Also why did this need to be spoilered, this is 2nd episode stuff.