# 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?

• 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 $L_0, L_1, L_2,\dots$.

Update (Jan. 12, 2016): I learned a few days ago from Ellen Eischen that the Kubota Tractor Corporation has a model called the "(compact) Standard L-Series," and today I saw a Kubota L-series go past my department building. Here is a photo I took.

If you're looking for a modern reinterpretation of what the L stands for in L-series, the webpage https://www.kubota.com/product/tlbseries.aspx gives the answer, and it's not Langlands: L means Loader or Landscaper.

• Is it so far-fetched that the $L$ stands for Legendre? On page 317 in the referenced paper, in setting notation he notes that there is the special case of the Legendre symbol, and then he inserts this into his series. – John Voight Apr 23 '19 at 22:41
• @JohnVoight if he were going to use that letter specifically for Legendre, I'd think it would be more because Legendre conjectured the theorem that Dirichlet was proving rather than because of the special case of Legendre symbols as a character mod p. We'll never know. – KConrad Apr 24 '19 at 3:48
• Yes, who knows! I read quickly a few other of Dirichlet's nearby papers, and I think I'm coming to your conclusion: he seems to use L just as a letter for a function, he wasn't trying to thoughtfully set notation, and it just stuck! – John Voight Apr 25 '19 at 13:30
• Yes, "L" for letter. :) – KConrad Apr 25 '19 at 14:36
• In French too, "lettre"! :) But I guess not in German, which is the language he was writing... – John Voight Apr 26 '19 at 15:22

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.

• 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'.