Compute the L-moments of the Kendall Function ($F_K(z; \mathbf{C})$) of a copula $\mathbf{C}$ where the $z$ is the joint probability of the $\mathbf{C}$. The Function is the cumulative distribution function (CDF) of the joint probability $Z$ of the coupla. The expected value of the $z(F_K)$ (mean, first L-moment $\lambda_1$), because $Z$ has nonzero probability for $0 \le Z \le \infty$, is$$\mathrm{E}[Z] = \lambda_1 = \int_0^\infty [1 - F_K(t)] = \int_0^1 [1 - F_K(t)] \,\mathrm{d}t\mbox{,}$$
where for circumstances here $0 \le Z \le 1$. The $\infty$ is mentioned only because expectations of such CDFs are usually shown using $(0,\infty)$ limits, whereas integration of quantile functions (CDF inverses) use limits $(0,1)$. Because the support of $Z$ is $(0,1)$, like the probability $F_K$, showing just it ($\infty$) as the upper limit could be confusing---statements such as ``probabilities of probabilities'' are rhetorically complex so pursuit of word precision is made herein.
An expression for $\lambda_r$ for $r \ge 2$ in terms of the $F_K(z)$ is
$$\lambda_r = \frac{1}{r}\sum_{j=0}^{r-2} (-1)^j {r-2 \choose j}{r \choose j+1} \int_{0}^{1} \! [F_K(t)]^{r-j-1}\times [1 - F_K(t)]^{j+1}\, \mathrm{d}t\mbox{,}$$
where because of these circumstances the limits of integration are $(0,1)$ and not $(-\infty, \infty)$ as in the usual definition of L-moments.
The mean, L-scale, coefficient of L-variation ($\tau_2$, LCV, L-scale/mean), L-skew ($\tau_3$, TAU3), L-kurtosis ($\tau_4$, TAU4), and $\tau_5$ (TAU5) are computed. In usual nomenclature, the L-moments are
$\lambda_1 = \mbox{mean,}$
$\lambda_2 = \mbox{L-scale,}$
$\lambda_3 = \mbox{third L-moment,}$
$\lambda_4 = \mbox{fourth L-moment, and}$
$\lambda_5 = \mbox{fifth L-moment,}$
whereas the L-moment ratios are
$\tau_2 = \lambda_2/\lambda_1 = \mbox{coefficient of L-variation, }$
$\tau_3 = \lambda_3/\lambda_2 = \mbox{L-skew, }$
$\tau_4 = \lambda_4/\lambda_2 = \mbox{L-kurtosis, and}$
$\tau_5 = \lambda_5/\lambda_2 = \mbox{not named.}$
It is common amongst practitioners to lump the L-moment ratios into the general term L-moments and remain inclusive of the L-moment ratios. For example, L-skew then is referred to as the 3rd L-moment when it technically is the 3rd L-moment ratio. There is no first L-moment ratio has no definition so results from this function will canoncially show a NA in that slot. The coefficient of L-variation is $\tau_2$ (subscript 2) and not Kendall's Tau ($\tau$). Sample L-moments are readily computed by several packages in R(e.g. lmomco, lmom, Lmoments, POT).