Expected number of returns simple random walk

Question

Let $(S_j)_{j \in \mathbb{N}}$ be a simple random walk on $\mathbb{Z}$ starting from the origin. It is well known that the expected number of returns to the origin after $n$ steps grows as $\sqrt{n}$, namely $$ E \Big ( \Big | \Big \{j \in \{0, \ldots, n\} \, : \, S_j = 0 \Big \} \Big | \Big )= c \sqrt{n} + o(\sqrt{n}), $$ where $E$ denotes the expectation and $c \in (0, \infty)$ is some constant. Are also the numerical value of the constant $c$ and the order of magnitude of the second term in the right-hand side of the previous expression known?