We study the behavior of zero-sets of the double zeta-function $$\zeta _2(s_1,s_2)$$ (and also of more general multiple zeta-function $$\zeta _r(s_1,\ldots ,s_r)$$). In our earlier paper we studied the case $$s_1=s_2$$, while in the present paper we consider a more general two-variables situation. We carry out numerical computations in order to trace the behavior of zero-sets of $$\zeta _2(s_1,s_2)$$. We observe that some zero-sets approach the points $$(s_1,s_2)$$ with $$s_2=0$$, while other zero-sets approach the points $$(s_1,s_2)$$ with $$s_2$$ being solutions of $$\zeta (s_2)=1$$. In the former case, when $$s_2$$ tends to 0, we observe that $$\mathrm{Im}\,s_1$$ comes close to the imaginary part of a non-trivial zero of the Riemann zeta-function. In the latter case we give a theoretical proof, in the general r-fold setting.