Let \(E=E(\sigma,\tau)\) be the compact selfadjoint positive definite operator on \(\ell^2(\mathbb{N})\) (whose matrix with respect to the standard basis is) given by \(E=\left[\frac{n^\sigma m^\sigma}{[n,m]^\tau}\right]_{n,m=1}^\infty\), where \([n,m]\) is the least common multiple (LCM) of \(n\) and \(m\) and \(\sigma,\tau\in\mathbb{R}\) satisfy \(\tau>0\), \(\tau>\sigma+1/2\) and \(\rho:=\tau-2\sigma>0\). By a prime factorization of \(n\) and \(m\), it follows that the eigenvalues of \(E\) can be written as products over primes \(p\) of eigenvalues of operators \(E_p=[p^{(j+k)\sigma} p^{-\tau\max(j,k)}]_{j,k=0}^\infty\). Asymptotic analysis of the eigenvalues of the latter, leads to the main result: \(\lambda_n(E)=\kappa n^{-\rho}+o(n^{-\rho})\) with \(\kappa=\kappa(\sigma,\tau)>0\). For \(\rho=1\) or \(\rho=1/2\), explicit values for \(\kappa\) can be obtained. Two applications are considered. Using the Riemann zeta function \(\zeta(s)\) a lower triangular Toeplitz operator \(T=T(\psi_\sigma)\) with symbol \(\psi_\sigma(s)=\zeta(\sigma+s)\) can be considered, and the spectral norm of a finite section \(T_N\) gives a lower bound for \(\max_{|t|\le T}|\zeta(\sigma+it)|\) for \(1/2<\sigma\le1\) as \(N=N(\sigma,T)\to\infty\). Since \(T_N^*T_N=\zeta(2\sigma)E(\sigma,2\sigma)\) this is linked to the previous results. Using a connection with the Beurling zeta function (see [H.G. Diamond and W.-B. Zhang, Beurling generalized numbers. Providence, RI: American Mathematical Society (AMS) (2016; Zbl 1378.11002)]), a sharper bound for the \(o(n^{-\rho})\) term in the main result can be obtained.