Judge Stephenson said that even though the officers’ accounts of the shooting differed from the video, that did not amount to proof that they were lying. “Two people with two different vantage points can witness the same event,” she said, and still describe it differently.

The judge suggested that key witnesses for the prosecution had offered conflicting testimony, and said there was nothing presented at trial that showed that the officers had failed to preserve evidence, as the prosecutors had argued. Challenging the point that officers had shooed away a witness as part of a cover-up, the judge said it was not obvious that the police had known the witness had seen the shooting.

The officers, who were brought to trial in November, were accused of writing in official reports that Laquan had tried to stab three other officers, saying they saw him trying to get up from the ground even after a barrage of shots.

Mr. March, Mr. Walsh and Mr. Gaffney each denied that they had conspired to come up with a narrative that might justify Mr. Van Dyke’s decision to shoot Laquan. None of them fired any shots that night. Other officers, too, had witnessed the shooting and had given questionable accounts, but were not on trial; grand jurors indicted the three officers but declined to indict any others.

[Read More: Was the Laquan McDonald case a turning point or an aberration?]

It was “undisputed and undeniable,” Judge Stephenson said, that Laquan had ignored officers’ commands to drop his knife. While she spoke, the three officers sat silently, sometimes staring down at the carpet or nervously jiggling a leg. After she read the verdict, several people broke into applause.