I should also add that my first reading of that word, going purely on the handwriting, would be Halle, only I have zero evidence either boy ever went there, much less when they were 18 and 20. So I tried really hard to make it Gottn., but that requires a bit of imagination. Without any other evidence for how this person forms their letters, I can't tell an 'e' from an 'n' or an 'a' from an 'o'*...but that is a pretty legit capital H (although as prinzsorgenfrei points out, anything goes with capitals) and no umlaut on the first vowel, and the t's aren't crossed (but then, they don't have to be)...so...Halle would have been my first reading.

It's also worth noting that I'm relying on a very unreliable secondary source for the sons' education (other than I know they went to Göttingen in 1760 because Lehndorff says so), and Halle is not out of the question.

* There are multiple ways of forming 'a'--this would be one of the less common ways, but not unprecedented.

TLDR: Halle or Göttn. take your pick!