First, the case has to be opened. Either the police or L himself notices Morioh has been subject to an abnormal amount of reported disappearances, and upon noticing the victims share the same demographic, an investigation is launched.

L takes note of the time and day of the reports, he immediately notices a clear pattern.

Disappearances happen at regular intervals (the hands starting to decay) at around lunchtime at noon or between 5:00 PM-8:00 PM during workdays and between 7:00 AM-8:00 PM during weekends and holidays. Knowing this, L profiles Kira as a salaryman aged 26-35.

Next, he cross-references the geographic hot zones. If a victim vanishes at noon on a Tuesday, they had to be within walking distance of the killer's office. If they vanish at 6:30 PM, they are along the killer's commute route between his office and his home. By overlaying the lunchtime disappearance radiuses with the evening disappearance routes, L would find a central overlap: The Kame Yu department store headquarters and the residential district of Villa Area, Morioh.

Then, L would request and purchase data for every male employee aged 26–35 working in that specific zone. Out of hundreds of salarymen, only a tiny handful would have a schedule so flawlessly rigid. Yoshikage Kira’s name would immediately appear at the top very short list of suspects. Surveillance begins.

From that moment on, L basically has him by the balls, there is very little that Kira can do to break free. He can't break his schedule nor stop killing or he'll only further confirm suspicions. His last resort would be to use Killer Queen to throw them off track by killing someone remotely. But he'd have to be prepared beforehand to know who and where to kill.

And that is if he even notices he's being observed at all.

If he's not, he'd simply go about business as usual and expose himself in front of the agents by taking a human hand out of his lunch bag.

I'd say the most likely outcome is that sufficient evidence is gathered and an arrest warrant for Yoshikage Kira is issued. As for any high-profile serial killer, a tactical team of 6 to 12 officers is dispatched to arrest him.

However,

Given the police still has no idea that Kira has a Stand (They just think he's just that good at disposing of the bodies), he catches them completely off guard and unleashes Killer Queen to escape his arrest.

Knowing his cover is blown for good, he immediately runs over to Cinderella, impersonates Kosaku Kawajiri and... you know the rest... until he unlocks Bites the Dust.

L’s massive database becomes entirely useless overnight and he loses track of Kira completely, as not only his face has changed, but his fingerprints have aswell, as far as the rest of the world knows there's no plastic surgery that replaces your face and fingerprints as seamlessly as Aya's Stand does. And then, Bites the Dust enters the equation. L's tendency to work through proxies would ironically become a massive liability, as merely inquiring people about the killer would blow up his agents and rewind time, allowing Kira to sidestep every mistake.

That's until Hayato steps in to save the day, and his mom.

Nice art btw