Let
G
G
be an FAb compact
p
p
-adic analytic group and suppose that
p
>
2
p>2
or
p
=
2
p=2
and
G
G
is uniform. We prove that there are natural numbers
n
1
,
…
,
n
k
n_1, \ldots , n_k
and functions
f
1
(
p
−
s
)
,
…
,
f
k
(
p
−
s
)
f_1(p^{-s}),\ldots , f_k(p^{-s})
rational in
p
−
s
p^{-s}
such that
\[
ζ
G
(
s
)
=
∑
λ
∈
Irr
(
G
)
λ
(
1
)
−
s
=
∑
i
=
1
k
n
i
−
s
f
i
(
p
−
s
)
.
\zeta ^G(s)=\sum _{\lambda \in \operatorname {Irr}(G)} \lambda (1) ^{-s}=\sum _{i=1}^kn_i^{-s}f_i(p^{-s}).
\]