Regularity properties of Minakshisundaram–Pleijel zeta function

Let $(M,g)$ be a closed (compact, no boundary) smooth $n$-dimensional Riemannian manifold. The Laplace–Beltrami operator $Delta_g$ on $M$ has discrete spectrum $(lambda_j)_j$ (indexed without multiplicity) with corresponding eigenfunctions $(psi_j)_j$, normalized in $L^2(rm{vol}_g)$. The MP zeta function of $Delta_g$ at a point $p$ in $M$ is defined as
$$zeta^{Delta_g}_p(s):=sum_j lambda_j^{-s} psi_j(p)^2, qquad sin mathbb C.$$

Question: Is something known about the continuity/differentiability of $pmapsto zeta^{Delta_g}_p(s)$ as a real-valued function on $M$, say, for real $s>n/2$ ?

As far as I understand, the original paper by Minakshisundaram and Pleijel is only concerned with properties of $zeta^{Delta_g}_p(s)$ as a meromorphic function of $s$, holomorphic for $Re(s)>n/2$. However, it seems to me that the authors do not discuss the properties of $pmapsto zeta^{Delta_g}_p(s)$ for fixed $s$ (equivalently, of the polynomials $A$ in Eqn. (28) in their paper).

Based on Hörmander-type estimates on the $L^infty$-norm of $psi_j$ and Weyl’s asymptotic, one can show that, for large enough $s$ (e.g. $sgeq n$), the function $pmapsto zeta^{Delta_g}_p(s)$ is continuous (by showing the total convergence of the series). This is however unsatisfactory, for I would hope that $pmapsto zeta^{Delta_g}_p(s)$ is continuous (possibly even smooth) for all $s>n/2$, which is the threshold originating in Weyl’s asymptotic for the $lambda_j$‘s. Is there any result in this direction which holds for all $s>n/2$?