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We give a certain sharp evaluation formula for the Ozsv\'ath-Szab\'o$d$-invariant of any Brieskorn homology sphere $\Sigma(p,q,r)$ with $p$ odd and$pq+pr-qr=1$. By using this result, we will present concrete examples that canbe calculated with elementary methods.

The $d$-invariant of any Brieskorn homology sphere with an equation