  There exist many explicit evaluations of Dirichlet series. Most of them are
constructed via the same approach: by taking products or powers of Dirichlet
series with a known Euler product representation. In this paper we derive a
result of a new flavour: we give the Dirichlet series representation to
solution $f=f(s,w)$ of the functional equation $L(s-wf)=\exp(f)$, where $L(s)$
is the L-function corresponding to a completely multiplicative function. Our
result seems to be a Dirichlet series analogue of the well known
Lagrange-B\"urmann formula for power series. The proof is probabilistic in
nature and is based on Kendall's identity, which arises in the fluctuation
theory of L\'evy processes.
