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In this paper, we consider a random entire function $f(s,\omega )$ defined by a random Dirichlet series ${\sum }_{n=1}^{\infty }{X}_n\left(\omega \right){\mathrm{e}}^{-{\lambda }_{n}s}$ where $X_n$ are independent and complex valued variables, $0\le {\lambda }_n\nearrow +\infty$. We prove that under natural conditions, for some random entire functions of order $\left(R\right)$ zero $f(s,\omega )$ almost surely every horizontal line is a Julia line without an exceptional value. The result improve a theorem of J. R. Yu: Julia lines of random Dirichlet series. Bull. Sci. Math. 128 (2004), 341–353, by relaxing condition on the distribution of $X_n$ for such function...