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Police have said Olango, 38, was behaving erratically and took a "shooting stance" immediately before he was killed by an officer. They later said Olango was holding an electronic cigarette at the time he was shot.
The incident has roiled the San Diego suburb, sparking days of demonstrations and protests, some of which turned violent and resulted in arrests.
On Wednesday, Sharpton emphasized that independent investigators must look into the shooting to avoid a conflict of interest.
"Here we are ... with people that left Uganda looking for freedom, looking for justice, looking for fairness and ... they had to bury their son," Sharpton said. "They had to deal with those that are sworn to protect us, harming us.
"All they've asked for ... is justice," he said. "Why is their son and brother dead?"
Civil rights leaders and Olango's relatives have questioned the El Cajon department's handling of the investigation. Some have criticized police for failing to make videos of the shooting public and choosing instead to release a single still image that appeared to implicate Olango.
The Rev. Shane Harris, head of the San Diego chapter of the National Action Network, said Wednesday that Olango's sister initially had called 911 three times. Police were subsequently dispatched on a 5150 call — requests for an involuntary psychiatric hold.
Once police arrived on the scene, authorities said, Olango ignored multiple orders and concealed his hand in his pants pockets before rapidly drawing out an object and placing both hands together on it in a shooting stance.
Officer Richard Gonsalves opened fire, while Officer Josh McDaniel shot Olango with a Taser, authorities said.
The image released by police showed Olango pointing at an officer's face as if he had a gun. After several nights of unrest in the city, El Cajon authorities released video recordings of the shooting. One was surveillance video from a nearby business, and the other had been taken by a witness using a cell phone.
Sharpton said Wednesday that he had seen video of the shooting, and while he said it appeared unlawful, he refused to "prejudge evidence."
"My question, though, after watching the video, is when do police shoot to kill and when do they shoot to wound?" he said.
Harris had been among those calling for the release of the videos. On Wednesday, he said Olango's death had "woke the whole world."
The press conference preceded a discussion of criminal justice issues at the National Action Network's Los Angeles headquarters. Several dozen community members, clergy members and civic leaders took part; the event was closed to the media.
Najee Ali, political director for the group's L.A. chapter, said the hour-long meeting was "emotionally charged" and that participants spent much of the time pledging their support to Olango's family.
Olango's father, Richard Olango, said Wednesday that his son's death "should be a turning point in justice."
"Police are meant to protect and are not meant to kill," the father said. "Military people are trained to kill. So do we have military people in our police department in America, or paramilitary police?"
4:55 p.m.: This article was updated to clarify that police released two videos; it also includes additional reaction and new information about Wednesday's community meeting.