The answer is literally a piece of string - it really depends on the number of trials, e.g. in a 100 spins the 'expected' longest losing run 3; in a 1,000 spins the 'expected' longest losing run is 5; in 10,000 the 'expected' losing run is 7; while in 100,000 the 'expected' losing run is 9.

As I said the answer is a piece of string and how long it is.