# Why are they called L-functions?

I was hoping to see this pop up on the recent big list question about etymology or terms and symbols. Since it has not, and I can't find an answer, I will ask:

What is the reason for the $L$ in $L$-function? I've read that the general use of the term cames from Dirichlet's $L$-functions $L(s,\chi).$ Was there any motivation behind Dirichlet's use or was it just an arbitary choice?

If so, is there any compelling reason that we keep this name other than tradition?

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Here's one way of looking at it. Dirichlet had to use some letter. He used L. Whatever he had used---would you have asked what the reason was? Why do number theorists use T for Hecke algebras? It's just what someone chose and it stuck. It might be no more than that... –  Kevin Buzzard Apr 14 '10 at 19:37
Did Dirichlet actually use L? –  François G. Dorais Apr 14 '10 at 21:34
François, Dirichlet absolutely used $L$. Look at his papers on primes in arithmetic progressions. –  KConrad Apr 15 '10 at 2:43
Kevin has an excellent point. I just wondered because L-functions are very important tools with a non-descriptive name. I like Paul's retroactive interpretation that L stands for Langlands. –  Jamie Weigandt Apr 16 '10 at 21:39

It is not known why Dirichlet denoted his functions with an $L$. Perhaps he chose $L$ for Legendre (I am not serious). The reason may be alphabetical. Just before $L$-functions are introduced in his 1837 paper on primes in arithmetic progression (Math. Werke vol. 1, 313--342), there are certain functions $G$ and $H$, and the letters $I, J$, and $K$ may not have seemed appropriate labels for a function.

While $L(s,\chi)$ and $L(\chi,s)$ are common notations for the $L$-function of a character $\chi$, neither decorated notation is due to Dirichlet; he simply wrote different $L$-functions as $L_0, L_1, L_2,\dots$.

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Many have suggested that it comes from "Lejeune", as in "Johann Peter Gustav Lejeune Dirichlet". I have never seen this properly sourced and have often wondered if the claim is legitimate.

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I have heard that as well, but I always thought it was mostly a humorous way to say "We have no idea where the terminology actually comes from". –  Pete L. Clark Apr 14 '10 at 22:56
I heard that Dirichlet had some disdain of his Walloon heritage. If so, I doubt that he would decide to "honor" the part of his name that most reflects that... –  François G. Dorais Apr 14 '10 at 23:59

Whatever the historical reasons are, I think it is a good thing to use the terminology 'L-function' because of Langlands's amazing contribution to the theory of automorphic forms. Moreover Langlands functorialities are stated in terms of the 'L-group'.

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