On regular variation of entire Dirichlet series

Consider an entire (absolutely convergent in $\mathbb{C}$) Dirichlet series $F$ with the exponents $\lambda_n$, i.e., of the form $F(s)=\sum_{n=0}^\infty a_ne^{s\lambda_n}$, and, for all $\sigma\in\mathbb{R}$, put $\mu(\sigma,F)=\max\{|a_n|e^{\sigma\lambda_n}:n\ge0\}$ and $M(\sigma,F)=\sup\{|F(s)|:\operatorname{Re}s=\sigma\}$. Previously, the first of the authors and M.M.~Sheremeta proved that if $\omega(\lambda)1$. In the present article we prove that the exponents of entire Dirichlet series with the same property can form an arbitrary sequence $\lambda=(\lambda_n)_{n=0}^\infty$ not satisfying $\omega(\lambda)