  We examine the localized mode and the transmission of plane waves across a
capacitive impurity of strength $\Delta$, in a 1D bi-inductive electrical
transmission line where the usual discrete Laplacian is replaced by a
fractional one characterized by a fractional exponent $s$. In the absence of
the impurity, the plane wave dispersion is computed in closed form in terms of
hypergeometric functions. It is observed that the bandwidth decreases steadily,
as $s$ decreases towards zero, reaching a minimum width at $s=0$. The localized
mode energy and spatial profiles are computed in close form v\`{i}a lattice
Green functions. The profiles show a remnant of the staggered-unstaggered
symmetry that is common in non-fractional chains. The width of the localized
mode decreases with decreasing $s$, becoming completely localized at the
impurity site at $s=0$. The transmission coefficient of plane waves across the
impurity is qualitatively similar to its non-fractional counterpart ($s=1$),
except at low $s$ values ($s\ll 1$). For a fixed exponent $s$, the transmission
decreases with increasing $\Delta$
