The fatal shooting of the immigrant day laborer on Sept. 5 set off three days of demonstrations in the streets.
"We want justice and we're not going to let this guy off the hook," said Ron Gochez of Union Del Barrio.
Police said the officer opened fire when Manuel Jaminez, 37, lunged at him with a knife.
"There was no knife in the hands of Manuel Jaminez when he was shot in the head by Officer Frank Hernandez," Gochez said.
Hernandez, who is on administrative leave, has been involved in two prior officer-involved shootings.
Beck said the investigation will be completed in a much shorter time than the usual eight months because of the strong community reaction.
Some activists are calling on the district attorney to file murder charges against the officer.
"That's not going to happen," Beck said. "We will present a completed investigation to the district attorney and he will make a decision at that point."
Beck promised a thorough and transparent investigation.
"If anybody thinks I won't prosecute a corrupt police officer then they haven't paid attention to my history," Beck said.
"The call by self styled 'community organizations' for an LAPD officer to be charged with murder for shooting a drunk man who had threatened both innocent civilians and police officers is ludicrous," the Los Angeles Police Protective League said.