Bill Allombert on Wed, 09 Sep 2015 16:17:17 +0200

 GP interface for computing L functions

Dear PARI developers,

We just added to master a new family of functions in GP to compute with general
L functions.

Some simple examples:

? lfun(1,2) \\ zeta(2) = Pi^2/6
%1 = 1.6449340668482264364724151666460251892
? lfun(-4,2) \\ L_chi(2) = Catalan with chi(n)=(-4/n)
%2 = 0.91596559417721901505460351493238411078
? K=nfinit(x^2+1); \\ Q(i)
? lfun(K,2) \\ zeta_K(2) = Catalan*Pi^2/6 with K=Q(i)
%4 = 1.5067030099229850308865650481820713960
? E=ellinit("11a1");
? lfun(E,1) \\ L_E(1)
%6 = 0.25384186085591068433775892335090946104
? ellL1(E)
%7 = 0.25384186085591068433775892335090946105
? lfunzeros(1,[10,30]) \\ zeros of zeta(1/2+I*t) with 10<=t<=30
%8 = [14.134725141734693790457251983562470271,21.022039638771554992628479593896902777,25.010857580145688763213790992562821819]
? E2=ellinit("5077a1");
? lfunorderzero(E2)
%10 = 3
? lfun(E2,1,3) \\ third derivative of L_E2 at 1
%11 = 10.391099400715804138751850510360917070
? ellanalyticrank(E2)
%12 = [3,10.391099400715804138751850510360917070]
? K2=bnfinit(x^2+23); B=bnrinit(K2,1,1);
? alpha=lfun([B,[1]],0,1) \\ L'(0) where L is the Hecke L function associated to one of
\\ the non trivial character of Cl(Q(sqrt(-23))
%13 = 0.28119957432296184651205076406787829979-2.3468559338982653182012529427009500020E-60*I
? algdep(exp(alpha),3)
%38 = x^3-x-1
? bnrL1(B)
%39 = [[1,0.28119957432296184651205076406787829979+0.E-38*I],[1,0.28119957432296184651205076406787829979+0.E-38*I],[0,-3/2]]

Below is the list of new functions:

lfun                lfundiv             lfunmfspec          lfunsymsqspec
lfunabelianrelinit  lfunetaquo          lfunmul             lfuntheta
lfunan              lfunhardy           lfunorderzero       lfunthetainit
lfuncheckfeq        lfuninit            lfunqf              lfunzeros
lfunconductor       lfunlambda          lfunrootres
lfuncreate          lfunmfpeters        lfunsymsq

Cheers,
The lfun team,
Bill, Henri, Karim and Pascal