Round 1:Bats,he has better gear and feats,though L is no noob at all.Finding L should be no problem to batman,he has the gear and access to databases that allow so,also L isn't that good at hiding,then using facial recognition and all that would be easy,for L he could go the Bane's route and deduce WHO is batman(though IMO he wouldn't be able to do this before Batman finds him),now finding him in the Batcave a place that was designed to not be found by people with A LOT more resources than L,would be VERY hard.
I really don't think Batman can match L's detective skills. Lawliet has an almost super human instinct for solving crimes, pinpointing exactly who and where the culprit is with pretty much no proof. He knew instantly that Light was Kira; the rest of the show was figuring out a way to prove it.
His deductive reasoning is straight up ridiculous, surmising not only Kira's exact location within Japan and the fact that he is a student, but also simply proving that Kira actually exists and even figuring out his identity within a matter of days. I think his creative and unorthodox investigation tactics give him an edge over the dark knight when it comes to detective work.
Here's the problem with round 1, it isn't that L is a better detective, it's that Bruce is an easier target. It'd be easier for L to deduce that Batman has to be rich, located within Gotham and probably has childhood trauma.
If this is a contest of who's a better detective, it's close but Bruce has the better feats, not to mention unlike a lot of other fictional detectives, Bats has a tendency to make inane leaps in logic. We also need to consider morals, as L used methods that Bruce wouldn't use as they may cause someone to die, it gives L a few more options than Bruce would have in solving crimes.
That aside, L has the advantage in scenario 1, in round 2 he gets his face smashed in.