Why didn't Hans shoot Shoshanna at the beginning of the film?

I think that it was just a whim on his part and that he thought her prospects for survival were slim to none anyway. Besides, if he had we wouldn't have a story.

Hans landa was the Jew hunter, not the Jew killer. I think he enjoyed hunting more than killing. He even said that he was proud of his nickname. Not to mention the distance that she was from it is unlikely he would be hit her especially with a German Luger.

I read somewhere his character wouldn't kill someone in the back like a coward etc. also she was kinda far
And he *did* enjoy the hunt more than the kill i guess

The main problem I have is that Landa and his men had ample time to get her before she broke out from under the house and ran away. They knew there was one that was still alive but they basically let her get away.

"And what about the sh*t weasels, the ones that blast out the basement door?" - Col. Curtis

Probably, yes, She looked to be at least 100 yards away by the time he was in a position to fire.

I asked the same question the last time I viewed the film. Check out the quality answers here:
https://moviechat.org/tt0361748/Inglourious-Basterds/58c7f4ba2214d80b5ced343d/Why-did-Landa-let-Shosanna-go-at-the-beginning

At that range he'd have to be a real Frederic Zoller to hit her.

As others have stated, hitting her running away from him at 50+ yards out would take an excellent/lucky pistol-shot, by the time he looked like he might shoot she was probably closer to 100 yards away. It would have diminished his reputation if he shot several times and missed her completely, which was likely.