Annihilators of Laurent coefficients of the complex power for normal crossing singularity

Let $f$ be a real-valued real analytic function defined on an open set of $\mathbb{R}^n$. Then the complex power $f_+^\lambda$ is defined as a distribution with a holomorphic parameter $\lambda$. We determine the annihilator (in the ring of differential operators) of each coefficient of the principal part of the Laurent expansion of $f_+^\lambda$ about $\lambda=-1$ in case $f=0$ has a normal crossing singularity.