Equivalent conditions for representing analytic functions by exponential series

Let $L(\lambda)$ be an entire function of exponential type with simple zeros $\lambda_1, \lambda_2,\dots$; let $\overline D$ be the smallest closed convex set which contains all of the singularities of the function which is associated with $L(\lambda)$ in the sense of Borel. In [1] there are necessary and sufficient conditions on $L(\lambda)$ under which a function $f(z)$ which is analytic in $\overline D$ can be represented in $D$ by a Dirichlet series with exponents $\lambda_1, \lambda_2,\dots$ We obtain new equivalent conditions on $L(\lambda)$.