  Twisted trilayer graphene hosts two moir\'e superlattices originating from
two interfaces between graphene layers. However, the system is generally
unstable to lattice relaxation at small twist angles and is expected to show a
significantly modified electronic band structure. In particular, a helical
trilayer graphene - whose two twisted angles have the same sign - provides an
attractive platform with a flat band isolated by large energy gaps near the
magic angle, but the interplay between the lattice and the electronic degrees
of freedom is not well understood. Here, we performed a large-scale molecular
dynamics simulation to study the lattice relaxation of helical trilayer
graphenes and evaluated their electronic spectra with a tight-binding model
calculation. The comparison of the electronic spectra both with and without the
lattice relaxation reveals how the lattice relaxation significantly modifies
the electronic spectra particularly near the charge neutrality point. We also
investigated the local density of states to visualize the spatially-varying
electronic spectra that accords with domain patterns of moir\'e lattice
stackings. We propose these characteristic spectral features in the electronic
degrees of freedom of a relaxed helical trilayer graphene to be confirmed by
scanning probe techniques, such as scanning single-electron transistors and
scanning tunneling microscopes.
