Improvements to Turing's Method

This paper refines the argument of Lehman by reducing the size of the constants in Turing's method. This improvement is given in Theorem 1 and scope for further improvements is also given. Analogous improvements to Dirichlet L-functions and Dedekind zeta-functions are also included.Comment: 21 pages, third edition: expanded to include sections on Dirichlet L-functions and Dedekind zeta-function