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Complex zeros of an incomplete Riemann zeta function and of the incomplete gamma function are calculated as functions of the upper limit X of the definition integrals. It becomes apparent that not all, but only some, of the zero trajectories of the incomplete Riemann zeta function join a zero of the Riemann zeta function 1(s) for X -+ C. The remaining trajectories at least in the region considered, approach the zero trajectories of the incomplete gamma function.