Hardy's function \(Z(t)\), sometimes referred to as the signed modulus, is defined by \[ Z(t) = \left(\pi^ {-it}{\Gamma({1\over 4}+{1\over 2}it)\over\Gamma({1\over 4}-{1\over 2}it)}\right)^ {{1\over 2}}\zeta({1\over 2}+it). \] Let \[ {\mathcal Z}(s) = \left(\pi^ {{1\over 2}-s}{\Gamma({s\over 2})\over\Gamma({1-s\over 2})}\right)^ {{1\over 2}}\zeta(s), \] so that \({\mathcal Z}(s)={\mathcal Z}(1-s)\) by the functional equation of the zeta-function. In this paper the zeros of \({\mathcal Z}'(s)\) are studied. In this important work he author shows that the nontrivial zeros of \({\mathcal Z}'(s)\) all lie in the strip \(| \Re s - {1\over 2}| <{15\over 2}\), and that if Riemann hypothesis is true all of the nontrivial zeros lie on the critical line. This is an improvement of Conrey and Ghosh's work [\textit{J. Conrey} and \textit{A. Ghosh}, J. Lond. Math. Soc. (2) 32, 193--202 (1985; Zbl 0582.10028)]. He also unconditionally improves the previous \(1.4\) (subject to RH) to \(\sqrt{{7723\over 3230}}\)