The T-adic L-function is a unversal L-function which
interpolates classical L-functions of all p-power order
exponential sums associated to a polynomial f(x) defined
over a finite field. We study its T-adic analytic properties
(analytic continuation and its T-adic Newton polygon).
The T-adic Newton polygon provides a uniform improvement
to previous results on p-adic Newton polygon of exponential sums
in the non-ordinary case. This is joint work with Chunlei Liu.