The Library of Congress isn’t sure, but they think that they have a Cooper’s Hawk (Accipter cooperii) currently in residence in the main reading room. (You’d think there’d be a copy of Roger Tory Peterson in there somewhere.)
They also don’t know how to catch it.
The preferred method of reducing raptors to possession is a device called a bal-chatri, a small wood or metal cage covered with loops of monofilament (in the old days, horsehair). You place a pigeon in the cage, drop the cage on a reading room table, and go away. The hawk goes for the pigeon and gets his feet entangled in the loops. You return and there’s your hawk.