Approximations of Euler's constant

Not to be confused with Approximations of Euler's number (e).

Euler's constant (sometimes called the Euler–Mascheroni constant) is a mathematical constant, usually denoted by the lowercase Greek letter gamma (γ), defined as the limiting difference between the harmonic series and the natural logarithm. The first 50 decimal digits of Euler's constant are 0.57721566490153286060651209008240243104215933593992. It is unknown if Euler's constant is rational or irrational.

This list is not exhaustive by any means; ultimately, there are infinite numbers, so this list can always be expanded.

Organization
This list will be organized the same way as the list of approximations of π.

2-5 digits

 * 4/7 = 0.571... (convergent of γ's continued fraction; first approximation by denominator to be accurate to 2 decimal digits)
 * 11/19 = 0.579... (convergent of γ's continued fraction; 2 decimal digits)
 * 15/26 = 0.5769... (convergent of γ's continued fraction; 2 decimal digits)
 * 26/45 = 0.5778... (first approximation by denominator to be accurate to 3 decimal digits)
 * 56/97 = 0.5773... (3 decimal digits)
 * 71/123 = 0.57724... (convergent of γ's continued fraction; first approximation by denominator to be accurate to 4 decimal digits)