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

A Probabilistic Angle on One Loop Scalar Integrals

Recasting the $N$-point one loop scalar integral as a probabilistic problem, allows the derivation of integral recurrence relations as well as exact analytical expressions in the most common cases. $\epsilon$ expansions are derived by writing a formula that relates an $N$-point function in decimal dimension to an $N$-point function in integer dimension. As an example, we give relations for the massive 5-point function in dimension $n=4-2\epsilon$, $n=6-2\epsilon$. The reduction of tensor integrals of rank 2 with $N=5$ is achieved showing the method's potential. Hypergeometric functions are not needed but only integration of arcsine function whose analytical continuation is well known.Comment: 35 pages, no figure