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Let $M$ be a compact hyperbolic $3$-manifold with volume $V$. Let $L$ be alink such that $M\setminus L$ is hyperbolic. For any hyperbolic link $L$ in$M$, in this article, we establish an upper bound of the length of an $n^{th}$shortest closed geodesic as a logarithmic function of $V$ in $M\setminus L$.Our works complement the work of Lakeland and Leininger \cite{Christopher} onthe upper bound of systole length.

Closed Geodesic Length Bounds of Hyperbolic Link Complements in Hyperbolic $3$-Manifolds