Which is, according to wiki: A serial killer is typically defined as a person who murders three or more people over a period of more than 30 days, with a "cooling off" period between each murder, and whose motivation for killing is largely based on psychological gratification.

That's exactly Light, except that he veers into spree killer occasionally.

Besides, the entire reason for L's involvement is arresting the serial killer, Kira.

Light's vigilantism is only a facade by which to make his murders more palatable, he has repeatedly and clearly demonstrated that he is willing to kill people who have committed no crimes (other than breaking the 'law of god' where Kira is the god in question, but that renders the situation moot) in pursuit of his objectives - which also don't have anything to do with law or justice.