Evaluating LL-functions with few known coefficients

We address the problem of evaluating an $L$-function when only a small number of its Dirichlet coefficients are known. We use the approximate functional equation in a new way and find that is possible to evaluate the $L$-function more precisely than one would expect from the standard approach. The method, however, requires considerably more computational effort to achieve a given accuracy than would be needed if more Dirichlet coefficients were available.Comment: 14 pages; Added a new section where we evaluate L(1/2 + 100 i, Delta) to 42 decimal places using no Dirichlet series coefficients at al

Real zeros of quadratic Dirichlet L-functions

We show that for a positive proportion of real primitive Dirichlet characters chi, the associated Dirichlet L-function L(s,chi) has no zeros on the positive real axis. Prior to this it was not known whether or not there were infinitely many L-functions (from any family) with no positive real zeros.Comment: 38 page