The question isn't:

"how many rolls do we need so that I have a >50% chance of seeing a streak of length L or longer",

or

"how many rolls do we need so that the expected longest streak is of length L".

The question is:

"we have seen N rolls, what is my chance of seeing a streak of length L or longer?"

Which is kind of the same thing, but in the opposite direction, and apparently much harder to calculate.