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In analytic number theory, the function

$$ L(s,\chi_m) = \sum_{n=1}^\infty \frac{\chi_m(n)}{n^s}. $$

is called the Dirichlet L-function and has many important uses in the study of prime numbers. In particular, if $\chi_1$ is the trivial Dirichlet character, we have the identity

$$ L(s,\chi_1)=\zeta(s). $$

Why were such functions called "L-functions? What is the functional significance of the letter L?

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It's the original notation used by Dirichlet. The reason why he chose L, without commenting on the choice, rather than some other letter is not known. Chances are there is not much of a reason, and he could just as well have chosen another letter.

Answers to the MathOverflow question Why are they called L-functions? mainly also assert this, but also present some theories, including:

  1. It just fitted in naturally with the rest of the notation of a paper.

  2. It is for Legendre.

  3. It is for Lejeune.

I rather doubt 3. The plausibility of 2. is based on the original context being closely linked to Legendre, but it's still a bit of a stretch (and presented rather tongue-in-cheek on MO).

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