Episode #1.1 - S1-E1
Continuity mistake: Luther goes to see Zoe and the house is an end terrace with a red door. Later he goes back and it's mid terrace with a black door.
Suggested correction: No, I don't think this is an error. I think the house with the red door is Luther and Zoe's marital home and the house with the black door is Mark's home.
That's what I thought, but how did he know where Mark lived?
I think it's a mistake because Riply is outside keeping watch on Zoe's house. At that point, Luther didn't know who Mark was, so how could he know where he lives?
Episode #1.5 - S1-E5
Continuity mistake: The position of Zoe's sweater on her right shoulder is constantly changing while Luther is holding her dead body near the end of the episode.
Episode #1.6 - S1-E6
Continuity mistake: During the fight with the shotgun Luther's hand is right at the end of the gun barrel. When the gun goes off and breaks the glass his hands are further down. (00:50:00)
Episode #1.6 - S1-E6
Continuity mistake: At Waterloo International station at the end of the episode, after Luther gets the shotgun off DCI Reed, in the ensuring conversation John Luther's collar on his coat goes from being up to down between shots. (00:51:00)
