In this paper, we improve the result of growth about Dirichlet series, and research properties of independent randorm variables {Z_n} under the condition, EZ_n=0,α>0, such that, 0<σ_n²=E|Z_n|²≤α²E|Z_n|²<+∞. We obtain that random series ∑_n=0^∞Z_nexp{-λ_nσ} and ∑_n=0^+∞σ_nexp{-λ_nσ} a.s have the same abscissa of convergence, the order of growth, type function, and every point in Res=0 is a.s. Borel point (without exceptional value) of a general random Dirichlet series of order ρ(1/σ) in right half plane.