  Using the framework of infinite Matrix Product States, the existence of an
\textit{anomalous} dynamical phase for the transverse-field Ising chain with
sufficiently long-range interactions was first reported in [J.~C.~Halimeh and
V.~Zauner-Stauber, arXiv:1610:02019], where it was shown that
\textit{anomalous} cusps arise in the Loschmidt-echo return rate for
sufficiently small quenches within the ferromagnetic phase. In this work we
further probe the nature of the anomalous phase through calculating the
corresponding Fisher-zero lines in the complex time plane. We find that these
Fisher-zero lines exhibit a qualitative difference in their behavior, where,
unlike in the case of the regular phase, some of them terminate before
intersecting the imaginary axis, indicating the existence of smooth peaks in
the return rate preceding the cusps. Additionally, we discuss in detail the
infinite Matrix Product State time-evolution method used to calculate Fisher
zeros and the Loschmidt-echo return rate using the Matrix Product State
transfer matrix. Our work sheds further light on the nature of the anomalous
phase in the long-range transverse-field Ising chain, while the numerical
treatment presented can be applied to more general quantum spin chains.
