Note: This question was reposted from Bitcoin Stack Exchange, where it received alike answers.
You can see a little background about this on this bitcointalk post by the late Hal Finney.
$\beta$ and $\lambda$ are the values on the secp256k1 curve such that: $$\begin{align} \lambda^3 &= 1 \mod N \\ \beta^3 &= 1 \mod P \\ \end{align}$$
As seen here, in hex, $N$ and $P$ are: $$\begin{align} N &= \mathtt{FFFFFFFF\ FFFFFFFF\ FFFFFFFF\ FFFFFFFE\ BAAEDCE6\ AF48A03B\ BFD25E8C\ D0364141} \\ P &= \mathtt{FFFFFFFF\ FFFFFFFF\ FFFFFFFF\ FFFFFFFF\ FFFFFFFF\ FFFFFFFF\ FFFFFFFE\ FFFFFC2F} \\ \end{align}$$
The actual values of lambda and beta are easily verifiable and are: $$\begin{align} \lambda &= \mathtt{5363ad4c\ c05c30e0\ a5261c02\ 8812645a\ 122e22ea\ 20816678\ df02967c\ 1b23bd72} \\ \beta &= \mathtt{7ae96a2b\ 657c0710\ 6e64479e\ ac3434e9\ 9cf04975\ 12f58995\ c1396c28\ 719501ee} \\ \end{align}$$
The question for me is, how do you derive this? Can someone show me step-by-step how you can figure out these values?