  We study Kato and Perrin-Riou's critical slope p-adic L-function attached to
an ordinary modular form, using the methods of our earlier work with Lei. We
show that it may be decomposed as a sum of two bounded measures multiplied by
explicit distributions depending only on the local properties of the modular
form at p. We use this decomposition to prove results on the zeros of the
p-adic L-function, and we show that our results match the behaviour observed in
examples calculated by Pollack and Stevens.
