  The chromatic polynomial of a graph $G$, denoted $P(G,m)$, is equal to the
number of proper $m$-colorings of $G$ for each $m \in \mathbb{N}$. In 1990,
Kostochka and Sidorenko introduced the list color function of graph $G$,
denoted $P_{\ell}(G,m)$, which is a list analogue of the chromatic polynomial.
The list color function threshold of $G$, denoted $\tau(G)$, is the smallest $k
\geq \chi(G)$ such that $P_{\ell}(G,m) = P(G,m)$ whenever $m \geq k$. It is
known that for every graph $G$, $\tau(G)$ is finite, and in fact, $\tau(G) \leq
(|E(G)|-1)/\ln(1+ \sqrt{2}) + 1$. It is also known that when $G$ is a cycle or
chordal graph, $G$ is enumeratively chromatic-choosable which means $\tau(G) =
\chi(G)$. A recent paper of Kaul et al. suggests that understanding the list
color function threshold of complete bipartite graphs is essential to the study
of the extremal behavior of $\tau$. In this paper we show that for any $n \geq
2$, $\tau(K_{2,n}) \leq \lceil (n+2.05)/1.24 \rceil$ which gives an improvement
on the general upper bound for $\tau(G)$ when $G = K_{2,n}$. We also develop
additional tools that allow us to show that $\tau(K_{2,3}) = \chi(K_{2,3})$ and
$\tau(K_{2,4}) = \tau(K_{2,5}) = 3$.
