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The modified Bessel function of the second kind $K_{i\nu}(x)$ of imaginaryorder for fixed $x>0$ possesses a countably infinite sequence of real zeros.Recently it has been shown that the $n$th zero behaves like $\nu_n\sim \pin/\log\,n$ as $n\to\infty$. In this note we determine a more precise estimatefor the bahaviour of these zeros for large $n$ by making use of the knownasymptotic expansion of $K_{i\nu}(x)$ for large $\nu$. Numerical results arepresented to illustrate the accuracy of the expansion obtained.

On the $\nu$-zeros of the modified Bessel function $K_{i\nu}(x)$ of positive argument