As a continuation of their previous papers [ibid. 24, No.5, 650-664 (1984;

Zbl 0568.65010); No.10, 1497-1513 (1984;

Zbl 0561.65011)], the authors discuss the computation of complex zeros of the Hankel functions

${H}_{\nu}^{\left(1\right)}\left(z\right)$ and

${H}_{\nu}^{\left(2\right)}\left(z\right)$ with real index

$\nu $. The method is due to Newton’s iteration, starting from a suitable asymptotic approximate value. First they discuss on the complex zeros of

${K}_{\nu}\left(z\right)$, and give several formulas concerning these functions. In the computation of their derivatives, they use asymptotic formulas containing Ai-function, and they remark the relation

$Ai\left(z\right)={\pi}^{-1}{(z/3)}^{1/2}$ ${K}_{1/3}(2{z}^{3/2}/3)$. They give tables of first 50 zeros of

${H}_{0}^{\left(1\right)}\left(z\right)$ and

${H}_{1}^{\left(1\right)}\left(z\right)$.