Analytic continuation of Bernoulli numbers: $B_{1/2}$?

Question

$B_0$ is 1, $B_1$ is $\pm 1/2$, and so on (I'm not going to list every single Bernoulli number). The Bernoulli numbers $B_n$ are defined for every integer $n \ge 0$. But what about $B_{1/2}$? And $B_i$? How can the Bernoulli numbers be extended beyond integer terms? So far the only analytic continuation of $B_n$ I could find was $-nζ(1-n)$, but for my purposes I cannot explicitly use the zeta function like this. Thoughts?