It is nearly 40 years old, one of a dwindling number of examples of its architectural period and was named for one of the country’s best-known lawmen of the 2oth century.

And yet there is little love for the J. Edgar Hoover FBI Building among historical preservationists, many of whom are still smarting from a very public and contentious debate to protect another D.C. building, the Third Church of Christ, Scientist, which they ultimately lost.

Like the church at 16th and I streets NW, the Hoover Building is one of the District’s last examples of Brutalist architecture, a modernist style of fortress-like buildings often fashioned out of massive amounts of concrete. The Hoover Building no longer fits the needs of the FBI, however, and the federal government is expected to begin seeking alternative locations for the agency soon.

The public doesn’t typically consider the Brutalist buildings historical — they consider them ugly. The Hoover Building in particular was named the ugliest building on Earth earlier this year by a travel Web site. And now that Penn Quarter has evolved into a posh residential community, neighbors of the building want it gone.

“Certainly the general public detests it,” said Tersh Boasberg, former longtime chairman of the D.C. Historic Preservation Review Board.

Completed in 1974, the building occupies an entire block between 9th and 10th streets NW, north of Pennsylvania Avenue NW and south of E Street NW. It offers 2.4 million square feet of gross space, but its usable space is much smaller and deteriorating. Its exterior — with its dry moat and endless security barriers — makes it particularly uninviting.

Rebecca Miller, is executive director of the D.C. Preservation League, which recently applied to have interior aspects of the Corcoran Gallery of Art and Union Station named architectural landmarks. She said the organization had not yet made a formal decision about whether to submit an application for the Hoover Building, but she doubted there was support for the move among members of her board.

“There’s not a lot of love for this building,” Miller said. “And it’s such a primary piece of real estate that saving the building would be a very difficult endeavor.”

Indeed, the group’s chairman, John D. Bellingham, president of Falls Church-based Monarc Construction, said there was unlikely to be a similarly fierce battle for the Hoover building as there was for the church.

“Is it a building that a lot of people like? Probably not. Is it a building that people will try to preserve? My instinct tells me probably not,” he said.

Boasberg said he didn’t expect to see an uprising of support for the building, either. “I would very much doubt it,” he said.

That does not mean preservationists are not watching how the building will be treated as the General Services Administration considers its future.

Someone other than the Preservation League could nominate the building as a landmark, and the Committee of 100, which frequently weighs in on historic preservation, will consider the Hoover Building according to trustee Loretta Neumann. “It’s definitely something we’ll want to think a lot about,” she said.

“The FBI building is exactly at the age where a lot of buildings tend to be lost because they are underappreciated,” said Tom Mayes, deputy general counsel at the National Trust for Historic Preservation.

A building’s popularity, Mayes cautioned, wasn’t enough reason to have it torn down. “We would at a minimum encourage the GSA to conduct a very careful review of the historic character of the building,” he said.