  The pseudo-Gamma function is a key tool introduced recently by Cheng and
Albeverio in the proof of \break the density hypothesis. This function is
doubly symmetric, which means that it is reflectively symmetric about the real
axis by the Schwarz principle, whereas it is also reflectively symmmetric about
the half line where the real part of the variable is equal to $\tfrac{1}{2}$.
In this article, we sharpen the estimate given in the proof of the density
hypothesis for this doubly symmetric pseudo-Gamma function on the real axis
near the symmetry center by taking a different approach from the way used in
the density hypothesis proof directly from the definition, reducing the error
caused by the fact that the difference of two pivotal parameters in the
definition of the pseudo-Gamma function is much larger than the difference of
the variables in this particular case.
