We analyze spectral properties of the Hilbert $L$-matrix $$left(frac{1}{max(m,n)+ u}right)_{m,n=0}^{infty}$$ regarded as an operator $L_{ u}$ acting on $ell^{2}(mathbb{N}_{0})$, for $ uinmathbb{R}$, $ u eq0,-1,-2,dots$. The approach is based on a spectral analysis of the inverse of $L_{ u}$, which is an unbounded Jacobi operator whose spectral properties are deducible in terms of the unit argument ${}_{3}F_{2}$-hypergeometric functions. In particular, we give answers to two open problems concerning the operator norm of $L_{ u}$ published by L. Bouthat and J. Mashreghi in [Oper. Matrices 15, No. 1 (2021), 47--58]. In addition, several general aspects concerning the definition of an $L$-operator, its positivity, and Fredholm determinants are also discussed.
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