The L-scale in the general case:
$$\large
\lambda_{2}^{(s, t)} =
\sqrt{\pi / 2}\ L(t + \frac 1 2,\ t + 1)$$
where L is the analytical continuation of the Lah number (i.e. written in terms of gamma functions).
For the symmetric case:
$$\large
\tau_{2k}^{(t, t)} =
\frac{4^{k+t}}{k}
\frac{
B(k + t + \frac 1 2,\ -\frac 1 2)
B(1 + t,\ 2 + t)
}{
B(k + t + \frac 3 2,\ \frac 1 2 - k)
B(2k + t,\ \frac 3 2 - k)
}$$
source: me
The L-scale in the general case:
where
Lis the analytical continuation of the Lah number (i.e. written in terms of gamma functions).For the symmetric case:
source: me