  Let K be a p-adic field. We explore Igusa's p-adic zeta function, which is
associated to a K-analytic function on an open and compact subset of K^n. First
we deduce a formula for an important coefficient in the Laurent series of this
meromorphic function at a candidate pole. Afterwards we use this formula to
determine all values less than -1/2 for n=2 and less than -1 for n=3 which
occur as the real part of a pole.
