Zeta = lfuncreate(1) lfun(Zeta,2) lfun(Zeta,0,1) lfun(Zeta,1) lfun(Zeta,1+x+O(x^10)) lfunzeros(Zeta,20) lfunlambda(Zeta,2) G=znstar(4,1); G.clgp Dir=lfuncreate([G,[1]]); lfunparams(Dir) lfunan(Dir,30) lfun(Dir,2) Catalan znconreyexp(G,[1]) lfun(Mod(3,4),2) Dedek = lfuncreate(x^2+1); lfunparams(Dedek) lfun(Dedek,2) lfuneuler(Dedek,3) zeta(2)*Catalan L=lfunmul(Zeta,Mod(3,4)); lfun(L,2) L2=lfundiv(Dedek,1); lfun(L2,2) lfuneuler(L2,3) bnf = bnfinit(a^2+23); bnr = bnrinit(bnf, 1); bnr.clgp Hecke = lfuncreate([bnr,[1]]); lfunparams(Hecke) lfuneuler(Hecke,5) z=lfun(Hecke,0,1) algdep(exp(z),3) bnr = bnrinit(bnf, 23); bnr.clgp ZL = lfuncreate([bnr,Mat(11)]); lfunparams(ZL) lfuneuler(ZL, 5) lfun(ZL,1) E = ellinit([0,-1,1,-10,-20]); lfun(E,1) lfuneuler(E,5) ellbsd(E) lfunparams(E) L = lfunsympow(E,2); lfunparams(L) nf = nfinit(a^2-5); phi = (1+a)/2; E = ellinit([1,phi+1,phi,phi,0],nf); E.j E.disc N = ellglobalred(E)[1] tor = elltors(E) \\ Z/8Z om = E.omega per = om[1][1]*om[2][1]; tam = elltamagawa(E) bsd = (per*tam) / (tor[1]^2*sqrt(abs(nf.disc))) ellbsd(E) L1 = lfun(E,1) E = ellinit([0,-1,1,-10,-20]); L=lfuntwist(E,Mod(2,5)); lfunan(E,10) lfunan(Mod(2,5),10) lfunan(L,10) nf=nfinit(polcyclo(5,'a)); E2=ellinit(E[1..5],nf); localbitprec(64); lfun(E2,2) L2=lfuntwist(E,Mod(4,5)); lfun(E,2)*lfun(L2,2)*norm(lfun(L,2)) L=lfungenus2([x^2+x,x^3+1]); lfunparams(L) lfun(L,1) lfunan(L,5) N = nfinit(x^6+108); G = galoisinit(N); [T,o] = galoischartable(G); T~ galoisconjclasses(G) L = lfunartin(N,G,T[,3],o); lfuncheckfeq(L) L[2..5] z = lfun(L,0,1) algdep(exp(z),3) bnr = bnrinit(bnfinit(a^2+a+1),6); lfunan([bnr,[1]],100)==lfunan(L,100) E=ellinit([0,-1,1,-10,-20]); \\ or ellinit("11a1") if elldata is available P=elldivpol(E,3) Q=polresultant(P,y^2-elldivpol(E,2)) R=nfsplitting(Q) N=nfinit(R); G=galoisinit(N); [T,o]=galoischartable(G); T~ o minpoly(Mod(y^3+y, polcyclo(o,y))) L = lfunartin(N,G,T[,3],o); L[2..5] lfuncheckfeq(L) dT = galoischardet(G,T[,3],o) dL = lfunartin(N,G,dT,o); dL[2..5] mf=mfinit([3267,1,-3],1); M=mfeigenbasis(mf); C=mfcoefs(M[3],100); mfembed(M[3],C)[2][2..-1]==lfunan(L,100) S = lfunan(L,1000); SE = lfunan(E,1000); Smod3 = round(real(S))+round(imag(S)/sqrt(2)); [(Smod3[i]-SE[i])%3|i<-[1..#Smod3],gcd(i,33)==1] bnf6=bnfinit(a^6-3*a^5+6*a^4+4*a^3+6*a^2-3*a+1); bnr6=bnrinit(bnf6,1); L1=lfuncreate([bnr6,[1]]); L1[2..5] bnf4=bnfinit(a^4-a^3+3*a^2+a-1); pr4 = idealprimedec(bnf4,3)[1]; bnr4=bnrinit(bnf4,[pr4,[0,1]]); L2=lfuncreate([bnr4,[1]]); L2[2..5] LL = lfundiv(L1,L2); LL[2..5] round(lfunan(L,1000)-lfunan(LL,1000),&e) e