I am reading through Washington's construction of $p$-adic $L$ functions in chapter $5$ of his book Cyclotomic Fields. Instead of $p$-adically interpolating as Kubota and Leopoldt originally did, Washington simply gives an explicit formula $$ L_p(s,\chi)=\frac{1}{F}\cdot\frac{1}{s-1}\sum_{a=1,p\nmid a}^F\chi(a)(\omega(a)^{-1}a)^{1-s}\sum_{j=0}^\infty\binom{1-s}{j}(B_j)(F/a)^j $$ where $\chi$ is a character of conductor $f$, $F$ is a multiple of $p$ and $f$, and $\omega$ is the Teichmuller character. I have been able to get through the construction fine, but am confused by his Theorem 5.12. Namely, he states $$ L_p(s,\chi)=a_0+a_1(s-1)+a_2(s-1)^2+\dots $$ with $|a_0|\leq1$ and $p|a_i$ for $i\geq1$.
In what sense does he mean that $p|a_i$? I am under the impression that this is taking place in $\mathbb{C}_p$, in which case saying $p|a_i$ is a trivial statement. Are the $a_i$ supposed to be integers, or $p$-adic integers? I feel as if I am missing something simple but I do not know what. Thanks in advance.