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We study asymptotic spectral properties of the generalized Hilbert $L$-matrix\[  L_{n}(\nu)=\left(\frac{1}{\max(i,j)+\nu}\right)_{i,j=0}^{n-1}, \] for largeorder $n$. First, for general $\nu\neq0,-1,-2,\dots$, we deduce the asymptoticdistribution of eigenvalues of $L_{n}(\nu)$ outside the origin. Second, for$\nu>0$, asymptotic formulas for small eigenvalues of $L_{n}(\nu)$ are derived.Third, in the classical case $\nu=1$, we also prove asymptotic formulas forlarge eigenvalues of $L_{n}\equiv L_{n}(1)$. I particular, we obtain anasymptotic expansion of $\|L_{n}\|$ improving Wilf's formula for the bestconstant in truncated Hardy's inequality.

Asymptotic spectral properties of the Hilbert $L$-matrix