$C=\frac{3}{\pi^2}\sum c(p,q,r) \frac{f(p)+f(q)-f(r)}{f(p)f(q)f(r)}$, where the sum is over normalized Markoff triples $(p,q,r)$, the positive-integer solutions of $p^2+q^2+r^2=3pqr$ written with $p\leq q\leq r$ [4], $c(p,q,r)=1$ except that $c(1,1,1)=c(1,1,2)=1/2$, and $f(t)=\log((3t+\sqrt{9t^2-4})/2)=\operatorname{arcosh}(3t/2)$ [2].

The decimal $0.18071704711507\ldots$ printed in [1] omits the digit $1$ after $0.180717$; after that digit is restored, the first wrong digit is the thirteenth decimal digit. OEIS A261613 records the corrected digits [2].

D. Zagier, On the number of Markoff numbers below a given bound, Mathematics of Computation 39 (1982), no. 160, 709-723. (doi) (MR)

OEIS A261613

OEIS A261613 b-file

Wikipedia: Markov number

Wikipedia: Markov spectrum

Lagrange numbers $L_m$ of the Markov spectrum —   the Lagrange numbers $L_m$ are indexed by Markoff numbers $m$, whose counting function $M(x)$ has leading constant $C$

Markov forms —   the quadratic forms attached to normalized Markoff triples

Markov quadratic irrationals —   the quadratic irrationals attached to normalized Markoff triples

Freiman's constant —   the left endpoint of Hall's ray in the Markov spectrum [5], whereas $C$ is a counting-function constant for Markoff numbers

Entries are of type: real number

How well the digits are known: assumed-bound

Table is complete: yes

How they were obtained:

The stored value is the 72-digit decimal recorded in [3]. OEIS says those digits were computed using Markoff numbers up to $10^{40}$; this table takes that bound on trust rather than deriving an independent tail estimate.

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The generator transcribes the digit string and checks its leading digits against Zagier's rapidly convergent sum over the 56 normalized Markoff triples with largest member $r\leq10^7$, whose count was checked by an independent enumeration.