How much is the rent weekly in London Zone 3?

There's no such thing as 'on average'. Firstly, the rent for a small bedsit will obviously be a fraction of that for a 10-bedroom mansion. Even within a particular size range like a 2-bedroom-flat, an upmarket one in good location and excellent condition will of course cost much more than a run-down one in a crappy location.

Secondly, 'Zone 3' covers a huge area, and places like Wimbledon or Hampstead are much more expensive than, say, Beckton or Tottenham.

First figure out what sort of property you mean, then figure out where in Zone 3 you're talking about, and then look up some properties on websites like rightmove.co.uk etc.

For what? A studio flat or a 5-bedroom house? How close to amenities?