  Let $F$ be an entire function represented by absolutely convergent for all
$z\in\mathbb{C}$ Dirichlet series of the form $ F(z) =
\sum\nolimits_{n=0}^{+\infty} a_{n}e^{z\lambda_{n}},$\ where a sequence
$(\lambda_n)$ such that $\lambda_n\in\mathbb{R}\ \ (n\geq0)$,
$\lambda_n\not=\lambda_k$ for any $n\not=k$ and $(\forall n\geq 0):\
0\leq\lambda_n<\beta:=\sup\{\lambda_j:\ j\geq0\}\leq +\infty.$ {Let $h$ be
non-decrease positive continuous function on $[0,+\infty)$ and $\Phi$ increase
positive continuous on $[0,+\infty)$ function.}
  In this paper we {find} the condition {on} $(\mu_n)$ and $(\lambda_n)$ {such
that} the relation $F(x+iy)=(1+o(1))a_{\nu(x, F)}e^{(x+iy)\lambda_{\nu(x, F)}}
$ holds as
  $x\to +\infty$\ outside some set $E$ of finite logarithmic $h$-measure
uniformly in $y\in\mathbb{R}$.
