  This work is a sequel of a previous work of one of the authors (Y.\^O), which
treated certain congruence relation between an elliptic Gauss sum and a
coefficient of power series expansion at the origin of the lemniscate sine
function. We extend the previous result (in \cite{O}) which concerned only for
non-vanishing elliptic Gauss sums. We give new congruence relations between
power series coefficients of the lemniscate cosine function, which hold if and
only if the corresponding elliptic Gauss sum vanishes.
