Prosecutors in New York came out guns blazing in their legal showdown against Luigi Mangione ... wasting little time getting eyeballs on the video they say shows Luigi gunning down a health care executive.

Luigi is in court right now for a pretrial hearing in his state murder case, and prosecutors just showed the infamous footage of UnitedHealthcare CEO Brian Thompson being blown away on a Manhattan sidewalk on December 4 last year. AP

Prosecutors say Luigi is the person in the video who whips out a firearm and shoots Thompson in the back.

Luigi's attorney, Karen Friedman Agnifilo , objected to the video being shown in court ... but the judge overruled her. The entire video wasn't shown, just the part where you see a hooded figure pull out the weapon. UnitedHealth Group

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