Police were sent to the bridge around 2 a.m. Saturday and found the boy, who was pronounced dead at a hospital. His name wasn't immediately released.
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The teen slipped and fell "when climbing upon one of the arches, in order to post, apparently, a social media broadcast," Moore said Tuesday during a Los Angeles Police Commission meeting.
Moore said added patrols will continue at the location to "counter such reckless actions."
"Tragically we see that location, while it has spawned a great deal of pride in Los Angeles, it has also unfortunately served as a backdrop now for tragedies such as this," Moore said at the meeting.
The bridge opened last July. The $588-million span, which replaced an 84-year-old Art Deco span, runs 3,500 feet over the concrete-lined L.A. River and connects downtown to the historic Eastside.
Police closed the bridge several times after it became a hotspot for street racing, graffiti and illegal takeovers soon after opening.
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In one instance, video captured a barber cutting a person's hair in the middle of the bridge as cars passed by.
A man was also shot and killed on a pedestrian ramp of the bridge in January during an unauthorized filming of a music video.
The Associated Press contributed to this report.