Let $η_1$, $η_2,\ldots$ be independent copies of a random variable $η$ with zero mean and finite variance which is bounded from the right, that is, $η\leq b$ almost surely for some $b>0$. Considering different types of the asymptotic behaviour of the probability $\mathbb{P}\{η\in[b-x,b]\}$ as $x\to 0+$ we derive precise tail asymptotics of the random Dirichlet series $\sum_{k\geq 1}k^{-α}η_k$ for $α\in (1/2, 1]$.