VERY dependent on the law enforcement agency. Most local law enforcement agencies require you to be a police officer first (I believe this to be necessary, as you really need a "street degree" before jumping into LE-related aviation missions), while SOME state agencies (MD) are starting to hire civilians. That being said, the civilians are usually former military pilots with PLENTY of OPERATIONAL time (they weren't just putzing around).
Another thing to keep in mind is that "minimums" are very subjective. For example, Phoenix PD mandates that one must spend the first three years of their career in patrol before specializing (aviation, SWAT, investigations, etc.) HOWEVER, the average wait to get on the "dream team" (aviation unit) is almost NINE years.
You also will not be flying immediately - I talked to a friend with LAPD, and he said that, to the best of his knowledge (he's SWAT, not aviation), you are an "observer" (left seat) for quite some time before becoming PIC.
From what I've heard, it seems like a great gig, albeit a very hard one to get into. Federal agencies also have flying positions, but they tend to also hire individuals with prior LE/MIL experience, in addition to graduate-level educations.
Good luck!
J.