Let P and \(Q \ne 0\) be real numbers such that \(P^2-4Q>0\) . The Lucas sequence \(\{U_n\}_{n \ge 0}\) is defined by the recurrence relation \(U_{n+2}=P U_{n+1}- QU_{n}\) \((n \ge 0)\) with \(U_0=0\) and \(U_1=1\) . Kamano introduced the Lucas zeta function \( \Phi ^{(P,Q)}(s):=\sum ^{\infty }_{n=1} \frac{1}{U_n^s}, \qquad \sigma =\hbox {Re} \ s>0, \) and proved that the Lucas zeta function \(\Phi ^{(P,Q)}(s)\) can be meromorphically continued to the whole s-plane except for some simple poles. Moreover, he discussed the values of \(\Phi ^{(P,-1)}(s)\) at negative integers and found that \(-m\) with \(m\ge 0\) , \(m \equiv 2 \ (\bmod \ 4)\) are its integral zeros. In this paper we provide its zero-free regions in the half-plane \(\sigma =\hbox {Re}\ s >0\) and consider the zero-free regions in the half-plane \(\sigma <0\) with the assumption \( Q=\pm 1\) . In the case \(Q=1\) , we derive that the whole negative real axis is the zero-free interval of \(\Phi ^{(P,1)}(s)\) for \(P\ge 3\) and that all the compact subsets of \(H_{-2,0} \cup H_{-4,-2} \cup H_{-6,-4}\) are the zero-free regions of \(\Phi ^{(P,1)}(s)\) if P is large enough, where \(H_{a,b}= \{ s= \sigma +it{:}\, a< \sigma < b\}.\) And in the other case \(Q=-1\) , we derive that \( [-1,0)\cup \mathop {\bigcup } \nolimits _{ n \in \mathbb {N_+}} [ -4n-1,-4n+1 ] \) is the zero-free interval of \(\Phi ^{(P,-1)}(s)\) for \(P\ge 2\) and that all the compact subsets of \(H_{-2,0} \cup H_{-4,-2}\) are the zero-free regions of \(\Phi ^{(P,-1)}(s)\) if P is large enough.
