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The value of the electron's spin is $\pm \frac{\hbar}{2}$.

In the paper where Pauli introduces his Pauli matrices he already knows the value of spin.

I'd imagine it was through the Ster-Gerlach experiment or through spectral line-splitting, but those would depend on the electron's intrinsic magnetic moment instead of spin, and the magnetic moment to spin ratio is twice the magnetic moment to angular momentum value that would be classically expected.

If that's how it is, then how was it discovered?

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    $\begingroup$ Are you asking about the 1/2 part (vs integral spin) or the exact magnitude in MKS units? $\endgroup$ – Carl Witthoft Apr 4 at 12:55
  • $\begingroup$ The 1/2 part, since it has the same magnetic moment as if it had spin 1 but g-factor 1. $\endgroup$ – user140323 Apr 5 at 2:30
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Possibly this article is what you're looking for. Partial quote follows; the article goes on to discuss the match between spin 1/2 and Zeeman effect, and so on.

  1. THE DISCOVERY OF ELECTRON SPIN In 1921, Compton (9) suggested that electron spin would be an essential ingredient in any reasonable explanation of bulk paramagnetism and ferromagnetism. Unfortunately, Compton’s proposal had almost no impact on his contemporaries; thus our story really begins in January 1925. At that time, quantum mechanics did not yet exist, and the generally accepted view of the atomic world was still classical, although it was supplemented by the partly successful but confusing quantization rules of the Bohr–Sommerfeld model. The anomalous Zeeman effect was a persistent puzzle that absorbed the attention of several physicists. Alfred Lande, who was in Tubingen, ¨ Germany, in 1925, had devised a useful and well-known semiempirical vector model to describe atomic angular momenta and the Zeeman effect (10), but how and why that model worked were a mystery. Wolfgang Pauli also thought very carefully about the anomalous Zeeman effect, and his deep insight and wide knowledge of atomic spectra led him to formulate the exclusion principle at the end of 1924 (11). Having already attracted attention in 1921 at age 21 for his brilliant monograph on special and general relativity (12), confident of his own extraordinary abilities, and highly critical of others’ work when it appeared superficial or na¨ıve, Pauli would play an important role in the history of electron spin. On January 7, 1925, the 20-year-old Kronig arrived in Tubingen on a traveling fellowship from ¨ Columbia University in New York to spend several weeks working with Lande, W. Gerlach, and E. Back. Lande greeted Kronig on his arrival, told him that Pauli would be visiting Tubingen the ¨ next day, and showed Kronig a letter from Pauli that he had just received. The letter emphasized that to understand the anomalous Zeeman effect, it would be necessary to endow the electron with a fourth quantum number in addition to the three it already possessed in the Bohr–Sommerfeld model, and that the fourth quantum number could take only two discrete values. On reading this letter, Kronig was immediately struck with inspiration, and that very afternoon he invented a concept of electron spin.
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  • $\begingroup$ It doesn't really go into much detail on the theory behind it. $\endgroup$ – user140323 Apr 5 at 2:30

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