If 1 omega 1R is an eigenvalue of a time-delay system for the delay tau(0) then 1 omega o is also an eigenvalue for the delays tau(k) := tau(0) + k2 pi/omega, for any k is an element of Z We investigate the sensitivity. periodicity and invariance properties of the root 1 omega for the case that 1 omega is a double eigenvalue for some tau(k) It turns out that under natural conditions (the condition that the root exhibits the completely regular splunng property if the delay is perturbed), the presence of a double imaginary root ico for some delay tau(0) implies that 1 omega is a simple root for the other delays tau(k), k not equal 0. Moreover, we show how to characterize the root locus around 1 omega. The entire local root locus picture can be completely determined from the square root splitting of the double root. We separate the general picture into two cases depending on the sign of a single scalar constant; the imaginary part of the first coefficient in the square root expansion of the double eigenvalue. (C) 2010 Elsevier Ltd All rights reserved.