With Airbnb’s New York operations in the crosshairs of the state Attorney General, the company fought back this week with a media event highlighting the value of its short-term apartment rentals to the city economy.

Company executives unveiled a new study, which estimates the benefit to the New York City at $632 million a year.

An untold number of those rentals are illegal, under a state law that bans using apartments as short-stay hotels. A recent court ruling determined that a short-term rental through Airbnb or other service is legit as long as the apartment’s legal tenant is also present — but otherwise Airbnb hosts who rent out their apartments are breaking the law.

While company executives are clearly worried, New York City’s roughly 13,500 Airbnb hosts shouldn’t be. Only a tiny fraction of hosts have been the subject of complaints to the Mayor’s Office of Special Enforcement, which investigates menaces to quality of life, and fewer still have actually been penalized.

Between July 4 and October 1 of this year, the office recorded 216 complaints about illegal short-term rentals — some of which tipsters linked to Airbnb postings. At those addresses, just 21 property owners have been cited so far by the Department of Buildings for violations covering that period.

During the same three months, some 167,000 guests in New York City used Airbnb.

As a matter of practice, the Office for Special Enforcement only investigates when it receives reports, a spokesman for the city indicated, which can be made through the city’s 311 hotline, or directly to the office.

We’ve mapped complaints received by the Office of Special Enforcement in between July 4 and October 1, 2013:

One dot = an address that was the subject of an illegal hotel report to the Office of Special Enforcement. Click on a dot to see details on the complaint. Where a location has received more than one report, only one is shown. The New York World has removed references to street and email addresses that appeared in the complaint reports.

An Airbnb spokesman suggested the relatively small number of complaints showed the rentals are welcomed in New York City. “The Airbnb community in New York is respectful and committed to making neighborhoods in all five boroughs even better places to live and visit,” the spokesman said in an email.

Airbnb’s detractors include New York City’s hotel industry, as well as elected officials and housing advocates who argue that Airbnb’s model encourages property owners to convert much needed affordable housing into fly-by-night hotels.

Last month, state Attorney General Eric Scheiderman subpoenaed the company for client information in order to recover what he charged were uncollected taxes. The company is fighting off the order in state Supreme Court.

As tourists (and sometimes bedbugs along for the trip) march through buildings hosting short-term rentals, why isn’t the Mayor’s Office of Special Enforcement hearing more complaints about apartments rented out through Airbnb and its ilk? Tenants are sometimes scared to denounce illegal hotels, said Tom Cayler, who is the chairman of the Illegal Hotel Committee for the West Side Neighborhood Alliance, an organization that advocates for affordable housing. Cayler said his organization has been helping tenants submit such complaints.

“They’re scared, if they’re a rent-stabilized tenant, that if the landlord finds out they’re the ones who complained about this illegal hotel then they’ll be targeted for harassment,” he said.

Enforcement of the law, Cayler added, “should be ramped up, there is no question in my mind.”

New York City represents Airbnb’s largest market wordwide – three times greater than its second market, Paris, an Airbnb official said at the report release.

With expectations, unconfirmed, that the company will eventually have an initial public offering, some observers have remarked Airbnb’s woes in New York could affect the company’s startup valuation, which is currently estimated at as much as $2.5 billion.

Other services frequently mentioned in the Office for Special Enforcement complaints alleging illegal rentals include Wimdu and Home Away.