John Derbyshire in his book <A HREF="http://tomlr.free.fr/Math%E9matiques/Fichiers%20Claude/Nombres/Derbyshire%20-%20Prime%20Obsession%20-%20Bernhard%20Riemann%20and%20the%20Greatest%20Unsolved%20Problem%20in%20MathematicsAAA.pdf">PRIME OBSESSION</A> says on page 343: 

> "I’ll round off with a complete calculation of $\pi(1,000,000)$, the
> number of primes up to one million, using Riemann’s formula -- not for
> the fun of it, though it is of course great fun, but to make some
> important points about the error term."

He finds that secondary terms contribute an error of -29.37378.     
**My question:** 
How many zeta zeros are needed to find it?