  We conjecture that the p-adic L-function of a non-trivial irreducible even
Artin character over a totally real field is non-zero at all non-zero integers.
This implies that a conjecture formulated by Coates and Lichtenbaum at negative
integers extends in a suitable way to all positive integers. We also state a
conjecture that for certain characters the Iwasawa series underlying the p-adic
L-series have no multiple roots except for those corresponding to the zero at
s=0 of the p-adic L-function. We provide some theoretical evidence for our
first conjecture, and prove both conjectures by means of computer calculations
for a large set of characters (and integers where appropriate) over the
rationals and over real quadratic fields, thus proving many instances of
conjectures by Coates and Lichtenbaum and by Schneider. The calculations and
the theoretical evidence also prove that certain p-adic regulators
corresponding to 1-dimensional characters for the rational numbers are units in
many cases. We also verify Gross' conjecture for the order of the zero of the
p-adic L-function at s=0 in many cases. We gather substantial statistical data
on the constant term of the underlying Iwasawa series, and propose a model for
its behaviour for certain characters.
