Let \(D (\Lambda, R)\) be a set of entire functions \(F\) expressible in the whole plane by the Dirichlet series of finite orders in the sense of Ritt where \(\Lambda = \{\lambda_n \}\) is a sequence of numbers such that \(0 < \lambda_n \uparrow \infty \) and \[ \limsup_{n \to \infty} \frac{n}{\lambda_n}. \] The authors have investigated the asymptotic behavior of the function from \(D (\Lambda, R)\) on a curve going to infinity (Theorem 1). Moreover for some class of exponents they have obtained the criterium of the equivalence between the logarithm of the maximal term and the logarithm of the absolute value of the Dirichlet series on at least one unbounded sequence of points of the curve.