# Properties

 Label 24-252e12-1.1-c8e12-0-0 Degree $24$ Conductor $6.559\times 10^{28}$ Sign $1$ Analytic cond. $1.37020\times 10^{24}$ Root an. cond. $10.1320$ Motivic weight $8$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 285·5-s + 198·7-s + 1.79e4·11-s + 2.05e5·17-s + 7.43e4·19-s + 6.28e4·23-s − 6.91e5·25-s + 5.75e5·29-s + 1.44e6·31-s − 5.64e4·35-s − 2.05e6·37-s + 7.72e6·43-s − 1.20e7·47-s − 8.46e6·49-s + 5.50e6·53-s − 5.10e6·55-s − 7.51e6·59-s − 3.72e7·61-s − 3.68e7·67-s + 3.00e7·71-s + 9.50e7·73-s + 3.54e6·77-s + 8.51e6·79-s − 5.86e7·85-s − 8.30e7·89-s − 2.11e7·95-s − 6.72e8·101-s + ⋯
 L(s)  = 1 − 0.455·5-s + 0.0824·7-s + 1.22·11-s + 2.46·17-s + 0.570·19-s + 0.224·23-s − 1.77·25-s + 0.813·29-s + 1.56·31-s − 0.0376·35-s − 1.09·37-s + 2.25·43-s − 2.47·47-s − 1.46·49-s + 0.697·53-s − 0.558·55-s − 0.619·59-s − 2.68·61-s − 1.82·67-s + 1.18·71-s + 3.34·73-s + 0.100·77-s + 0.218·79-s − 1.12·85-s − 1.32·89-s − 0.260·95-s − 6.46·101-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+4)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$24$$ Conductor: $$2^{24} \cdot 3^{24} \cdot 7^{12}$$ Sign: $1$ Analytic conductor: $$1.37020\times 10^{24}$$ Root analytic conductor: $$10.1320$$ Motivic weight: $$8$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{252} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(24,\ 2^{24} \cdot 3^{24} \cdot 7^{12} ,\ ( \ : [4]^{12} ),\ 1 )$$

## Particular Values

 $$L(\frac{9}{2})$$ $$\approx$$ $$3.108273081$$ $$L(\frac12)$$ $$\approx$$ $$3.108273081$$ $$L(5)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
7 $$1 - 198 T + 8501949 T^{2} + 4349076514 p T^{3} + 168809648166 p^{3} T^{4} + 1246332270114 p^{6} T^{5} + 2613116425173 p^{10} T^{6} + 1246332270114 p^{14} T^{7} + 168809648166 p^{19} T^{8} + 4349076514 p^{25} T^{9} + 8501949 p^{32} T^{10} - 198 p^{40} T^{11} + p^{48} T^{12}$$
good5 $$1 + 57 p T + 773148 T^{2} + 42526161 p T^{3} + 382895712834 T^{4} + 210666306118233 T^{5} + 84509030530123142 T^{6} + 5383721618415903483 p T^{7} -$$$$10\!\cdots\!46$$$$p^{2} T^{8} -$$$$32\!\cdots\!89$$$$p^{4} T^{9} -$$$$34\!\cdots\!96$$$$p^{4} T^{10} -$$$$15\!\cdots\!89$$$$p^{6} T^{11} -$$$$90\!\cdots\!46$$$$p^{6} T^{12} -$$$$15\!\cdots\!89$$$$p^{14} T^{13} -$$$$34\!\cdots\!96$$$$p^{20} T^{14} -$$$$32\!\cdots\!89$$$$p^{28} T^{15} -$$$$10\!\cdots\!46$$$$p^{34} T^{16} + 5383721618415903483 p^{41} T^{17} + 84509030530123142 p^{48} T^{18} + 210666306118233 p^{56} T^{19} + 382895712834 p^{64} T^{20} + 42526161 p^{73} T^{21} + 773148 p^{80} T^{22} + 57 p^{89} T^{23} + p^{96} T^{24}$$
11 $$1 - 1629 p T - 228436530 T^{2} + 9086672730411 T^{3} - 91396129870216860 T^{4} -$$$$34\!\cdots\!27$$$$T^{5} +$$$$14\!\cdots\!18$$$$T^{6} -$$$$25\!\cdots\!13$$$$T^{7} +$$$$54\!\cdots\!48$$$$T^{8} -$$$$87\!\cdots\!79$$$$T^{9} +$$$$35\!\cdots\!14$$$$T^{10} +$$$$22\!\cdots\!59$$$$T^{11} -$$$$53\!\cdots\!22$$$$T^{12} +$$$$22\!\cdots\!59$$$$p^{8} T^{13} +$$$$35\!\cdots\!14$$$$p^{16} T^{14} -$$$$87\!\cdots\!79$$$$p^{24} T^{15} +$$$$54\!\cdots\!48$$$$p^{32} T^{16} -$$$$25\!\cdots\!13$$$$p^{40} T^{17} +$$$$14\!\cdots\!18$$$$p^{48} T^{18} -$$$$34\!\cdots\!27$$$$p^{56} T^{19} - 91396129870216860 p^{64} T^{20} + 9086672730411 p^{72} T^{21} - 228436530 p^{80} T^{22} - 1629 p^{89} T^{23} + p^{96} T^{24}$$
13 $$1 - 4680855171 T^{2} + 11303530637810821443 T^{4} -$$$$18\!\cdots\!56$$$$T^{6} +$$$$24\!\cdots\!01$$$$T^{8} -$$$$26\!\cdots\!25$$$$T^{10} +$$$$23\!\cdots\!98$$$$T^{12} -$$$$26\!\cdots\!25$$$$p^{16} T^{14} +$$$$24\!\cdots\!01$$$$p^{32} T^{16} -$$$$18\!\cdots\!56$$$$p^{48} T^{18} + 11303530637810821443 p^{64} T^{20} - 4680855171 p^{80} T^{22} + p^{96} T^{24}$$
17 $$1 - 205782 T + 32244830382 T^{2} - 3730708280511468 T^{3} +$$$$28\!\cdots\!73$$$$T^{4} -$$$$12\!\cdots\!92$$$$T^{5} -$$$$42\!\cdots\!50$$$$T^{6} +$$$$16\!\cdots\!58$$$$T^{7} -$$$$15\!\cdots\!06$$$$T^{8} +$$$$80\!\cdots\!30$$$$T^{9} +$$$$26\!\cdots\!82$$$$T^{10} -$$$$99\!\cdots\!60$$$$T^{11} +$$$$10\!\cdots\!69$$$$T^{12} -$$$$99\!\cdots\!60$$$$p^{8} T^{13} +$$$$26\!\cdots\!82$$$$p^{16} T^{14} +$$$$80\!\cdots\!30$$$$p^{24} T^{15} -$$$$15\!\cdots\!06$$$$p^{32} T^{16} +$$$$16\!\cdots\!58$$$$p^{40} T^{17} -$$$$42\!\cdots\!50$$$$p^{48} T^{18} -$$$$12\!\cdots\!92$$$$p^{56} T^{19} +$$$$28\!\cdots\!73$$$$p^{64} T^{20} - 3730708280511468 p^{72} T^{21} + 32244830382 p^{80} T^{22} - 205782 p^{88} T^{23} + p^{96} T^{24}$$
19 $$1 - 74313 T + 42805567800 T^{2} - 3044214245327301 T^{3} +$$$$64\!\cdots\!10$$$$T^{4} +$$$$13\!\cdots\!11$$$$T^{5} -$$$$20\!\cdots\!94$$$$T^{6} +$$$$16\!\cdots\!05$$$$T^{7} -$$$$13\!\cdots\!38$$$$T^{8} +$$$$11\!\cdots\!99$$$$p T^{9} +$$$$19\!\cdots\!48$$$$T^{10} -$$$$94\!\cdots\!83$$$$T^{11} +$$$$96\!\cdots\!66$$$$T^{12} -$$$$94\!\cdots\!83$$$$p^{8} T^{13} +$$$$19\!\cdots\!48$$$$p^{16} T^{14} +$$$$11\!\cdots\!99$$$$p^{25} T^{15} -$$$$13\!\cdots\!38$$$$p^{32} T^{16} +$$$$16\!\cdots\!05$$$$p^{40} T^{17} -$$$$20\!\cdots\!94$$$$p^{48} T^{18} +$$$$13\!\cdots\!11$$$$p^{56} T^{19} +$$$$64\!\cdots\!10$$$$p^{64} T^{20} - 3044214245327301 p^{72} T^{21} + 42805567800 p^{80} T^{22} - 74313 p^{88} T^{23} + p^{96} T^{24}$$
23 $$1 - 62832 T - 267130501926 T^{2} + 2408529172043712 T^{3} +$$$$34\!\cdots\!41$$$$T^{4} +$$$$78\!\cdots\!16$$$$T^{5} -$$$$30\!\cdots\!30$$$$T^{6} +$$$$53\!\cdots\!08$$$$T^{7} +$$$$24\!\cdots\!18$$$$T^{8} -$$$$18\!\cdots\!88$$$$T^{9} -$$$$23\!\cdots\!78$$$$T^{10} +$$$$86\!\cdots\!16$$$$T^{11} +$$$$21\!\cdots\!93$$$$T^{12} +$$$$86\!\cdots\!16$$$$p^{8} T^{13} -$$$$23\!\cdots\!78$$$$p^{16} T^{14} -$$$$18\!\cdots\!88$$$$p^{24} T^{15} +$$$$24\!\cdots\!18$$$$p^{32} T^{16} +$$$$53\!\cdots\!08$$$$p^{40} T^{17} -$$$$30\!\cdots\!30$$$$p^{48} T^{18} +$$$$78\!\cdots\!16$$$$p^{56} T^{19} +$$$$34\!\cdots\!41$$$$p^{64} T^{20} + 2408529172043712 p^{72} T^{21} - 267130501926 p^{80} T^{22} - 62832 p^{88} T^{23} + p^{96} T^{24}$$
29 $$( 1 - 287727 T + 1872284339769 T^{2} - 775009483994138304 T^{3} +$$$$17\!\cdots\!47$$$$T^{4} -$$$$75\!\cdots\!97$$$$T^{5} +$$$$10\!\cdots\!98$$$$T^{6} -$$$$75\!\cdots\!97$$$$p^{8} T^{7} +$$$$17\!\cdots\!47$$$$p^{16} T^{8} - 775009483994138304 p^{24} T^{9} + 1872284339769 p^{32} T^{10} - 287727 p^{40} T^{11} + p^{48} T^{12} )^{2}$$
31 $$1 - 1442952 T + 2753998863693 T^{2} - 2972426343990906600 T^{3} +$$$$45\!\cdots\!47$$$$T^{4} -$$$$43\!\cdots\!68$$$$T^{5} +$$$$45\!\cdots\!68$$$$T^{6} -$$$$38\!\cdots\!56$$$$T^{7} +$$$$33\!\cdots\!61$$$$T^{8} -$$$$81\!\cdots\!00$$$$p T^{9} +$$$$14\!\cdots\!59$$$$T^{10} -$$$$44\!\cdots\!44$$$$p T^{11} +$$$$91\!\cdots\!98$$$$T^{12} -$$$$44\!\cdots\!44$$$$p^{9} T^{13} +$$$$14\!\cdots\!59$$$$p^{16} T^{14} -$$$$81\!\cdots\!00$$$$p^{25} T^{15} +$$$$33\!\cdots\!61$$$$p^{32} T^{16} -$$$$38\!\cdots\!56$$$$p^{40} T^{17} +$$$$45\!\cdots\!68$$$$p^{48} T^{18} -$$$$43\!\cdots\!68$$$$p^{56} T^{19} +$$$$45\!\cdots\!47$$$$p^{64} T^{20} - 2972426343990906600 p^{72} T^{21} + 2753998863693 p^{80} T^{22} - 1442952 p^{88} T^{23} + p^{96} T^{24}$$
37 $$1 + 2058621 T - 12851151325950 T^{2} - 24300204894085231549 T^{3} +$$$$96\!\cdots\!96$$$$T^{4} +$$$$14\!\cdots\!17$$$$T^{5} -$$$$58\!\cdots\!58$$$$T^{6} -$$$$55\!\cdots\!85$$$$T^{7} +$$$$31\!\cdots\!92$$$$T^{8} +$$$$14\!\cdots\!77$$$$T^{9} -$$$$14\!\cdots\!02$$$$T^{10} -$$$$19\!\cdots\!77$$$$T^{11} +$$$$54\!\cdots\!90$$$$T^{12} -$$$$19\!\cdots\!77$$$$p^{8} T^{13} -$$$$14\!\cdots\!02$$$$p^{16} T^{14} +$$$$14\!\cdots\!77$$$$p^{24} T^{15} +$$$$31\!\cdots\!92$$$$p^{32} T^{16} -$$$$55\!\cdots\!85$$$$p^{40} T^{17} -$$$$58\!\cdots\!58$$$$p^{48} T^{18} +$$$$14\!\cdots\!17$$$$p^{56} T^{19} +$$$$96\!\cdots\!96$$$$p^{64} T^{20} - 24300204894085231549 p^{72} T^{21} - 12851151325950 p^{80} T^{22} + 2058621 p^{88} T^{23} + p^{96} T^{24}$$
41 $$1 - 51768779647524 T^{2} +$$$$11\!\cdots\!34$$$$T^{4} -$$$$14\!\cdots\!44$$$$T^{6} +$$$$95\!\cdots\!15$$$$T^{8} -$$$$23\!\cdots\!32$$$$T^{10} -$$$$50\!\cdots\!56$$$$T^{12} -$$$$23\!\cdots\!32$$$$p^{16} T^{14} +$$$$95\!\cdots\!15$$$$p^{32} T^{16} -$$$$14\!\cdots\!44$$$$p^{48} T^{18} +$$$$11\!\cdots\!34$$$$p^{64} T^{20} - 51768779647524 p^{80} T^{22} + p^{96} T^{24}$$
43 $$( 1 - 3860661 T + 41440561775907 T^{2} -$$$$15\!\cdots\!68$$$$T^{3} +$$$$92\!\cdots\!09$$$$T^{4} -$$$$28\!\cdots\!79$$$$T^{5} +$$$$13\!\cdots\!06$$$$T^{6} -$$$$28\!\cdots\!79$$$$p^{8} T^{7} +$$$$92\!\cdots\!09$$$$p^{16} T^{8} -$$$$15\!\cdots\!68$$$$p^{24} T^{9} + 41440561775907 p^{32} T^{10} - 3860661 p^{40} T^{11} + p^{48} T^{12} )^{2}$$
47 $$1 + 12088194 T + 98045514715926 T^{2} +$$$$59\!\cdots\!16$$$$T^{3} +$$$$26\!\cdots\!53$$$$T^{4} +$$$$10\!\cdots\!44$$$$T^{5} +$$$$28\!\cdots\!58$$$$T^{6} +$$$$14\!\cdots\!62$$$$T^{7} +$$$$11\!\cdots\!78$$$$T^{8} +$$$$80\!\cdots\!42$$$$T^{9} +$$$$49\!\cdots\!74$$$$T^{10} +$$$$26\!\cdots\!56$$$$T^{11} +$$$$13\!\cdots\!13$$$$T^{12} +$$$$26\!\cdots\!56$$$$p^{8} T^{13} +$$$$49\!\cdots\!74$$$$p^{16} T^{14} +$$$$80\!\cdots\!42$$$$p^{24} T^{15} +$$$$11\!\cdots\!78$$$$p^{32} T^{16} +$$$$14\!\cdots\!62$$$$p^{40} T^{17} +$$$$28\!\cdots\!58$$$$p^{48} T^{18} +$$$$10\!\cdots\!44$$$$p^{56} T^{19} +$$$$26\!\cdots\!53$$$$p^{64} T^{20} +$$$$59\!\cdots\!16$$$$p^{72} T^{21} + 98045514715926 p^{80} T^{22} + 12088194 p^{88} T^{23} + p^{96} T^{24}$$
53 $$1 - 5506743 T - 157455075543936 T^{2} +$$$$13\!\cdots\!05$$$$T^{3} +$$$$90\!\cdots\!54$$$$T^{4} -$$$$26\!\cdots\!59$$$$p T^{5} -$$$$12\!\cdots\!18$$$$T^{6} +$$$$10\!\cdots\!63$$$$T^{7} -$$$$26\!\cdots\!82$$$$T^{8} -$$$$55\!\cdots\!41$$$$T^{9} +$$$$34\!\cdots\!56$$$$T^{10} +$$$$14\!\cdots\!19$$$$T^{11} -$$$$26\!\cdots\!58$$$$T^{12} +$$$$14\!\cdots\!19$$$$p^{8} T^{13} +$$$$34\!\cdots\!56$$$$p^{16} T^{14} -$$$$55\!\cdots\!41$$$$p^{24} T^{15} -$$$$26\!\cdots\!82$$$$p^{32} T^{16} +$$$$10\!\cdots\!63$$$$p^{40} T^{17} -$$$$12\!\cdots\!18$$$$p^{48} T^{18} -$$$$26\!\cdots\!59$$$$p^{57} T^{19} +$$$$90\!\cdots\!54$$$$p^{64} T^{20} +$$$$13\!\cdots\!05$$$$p^{72} T^{21} - 157455075543936 p^{80} T^{22} - 5506743 p^{88} T^{23} + p^{96} T^{24}$$
59 $$1 + 7511901 T + 219421003168794 T^{2} +$$$$15\!\cdots\!27$$$$T^{3} +$$$$11\!\cdots\!40$$$$T^{4} +$$$$22\!\cdots\!01$$$$T^{5} +$$$$33\!\cdots\!86$$$$T^{6} +$$$$21\!\cdots\!59$$$$T^{7} +$$$$96\!\cdots\!08$$$$T^{8} -$$$$51\!\cdots\!75$$$$T^{9} -$$$$12\!\cdots\!74$$$$T^{10} -$$$$51\!\cdots\!05$$$$T^{11} -$$$$15\!\cdots\!70$$$$T^{12} -$$$$51\!\cdots\!05$$$$p^{8} T^{13} -$$$$12\!\cdots\!74$$$$p^{16} T^{14} -$$$$51\!\cdots\!75$$$$p^{24} T^{15} +$$$$96\!\cdots\!08$$$$p^{32} T^{16} +$$$$21\!\cdots\!59$$$$p^{40} T^{17} +$$$$33\!\cdots\!86$$$$p^{48} T^{18} +$$$$22\!\cdots\!01$$$$p^{56} T^{19} +$$$$11\!\cdots\!40$$$$p^{64} T^{20} +$$$$15\!\cdots\!27$$$$p^{72} T^{21} + 219421003168794 p^{80} T^{22} + 7511901 p^{88} T^{23} + p^{96} T^{24}$$
61 $$1 + 37215576 T + 801351511399062 T^{2} +$$$$12\!\cdots\!20$$$$T^{3} +$$$$13\!\cdots\!93$$$$T^{4} +$$$$13\!\cdots\!60$$$$p T^{5} -$$$$12\!\cdots\!50$$$$T^{6} -$$$$38\!\cdots\!92$$$$T^{7} -$$$$59\!\cdots\!10$$$$T^{8} -$$$$64\!\cdots\!36$$$$T^{9} -$$$$41\!\cdots\!10$$$$T^{10} +$$$$10\!\cdots\!04$$$$T^{11} +$$$$74\!\cdots\!73$$$$T^{12} +$$$$10\!\cdots\!04$$$$p^{8} T^{13} -$$$$41\!\cdots\!10$$$$p^{16} T^{14} -$$$$64\!\cdots\!36$$$$p^{24} T^{15} -$$$$59\!\cdots\!10$$$$p^{32} T^{16} -$$$$38\!\cdots\!92$$$$p^{40} T^{17} -$$$$12\!\cdots\!50$$$$p^{48} T^{18} +$$$$13\!\cdots\!60$$$$p^{57} T^{19} +$$$$13\!\cdots\!93$$$$p^{64} T^{20} +$$$$12\!\cdots\!20$$$$p^{72} T^{21} + 801351511399062 p^{80} T^{22} + 37215576 p^{88} T^{23} + p^{96} T^{24}$$
67 $$1 + 36824553 T + 235041725822940 T^{2} +$$$$20\!\cdots\!73$$$$T^{3} +$$$$97\!\cdots\!62$$$$T^{4} +$$$$96\!\cdots\!49$$$$T^{5} +$$$$10\!\cdots\!34$$$$T^{6} +$$$$80\!\cdots\!19$$$$T^{7} +$$$$10\!\cdots\!94$$$$T^{8} -$$$$43\!\cdots\!05$$$$T^{9} +$$$$18\!\cdots\!76$$$$T^{10} +$$$$28\!\cdots\!67$$$$T^{11} -$$$$88\!\cdots\!26$$$$T^{12} +$$$$28\!\cdots\!67$$$$p^{8} T^{13} +$$$$18\!\cdots\!76$$$$p^{16} T^{14} -$$$$43\!\cdots\!05$$$$p^{24} T^{15} +$$$$10\!\cdots\!94$$$$p^{32} T^{16} +$$$$80\!\cdots\!19$$$$p^{40} T^{17} +$$$$10\!\cdots\!34$$$$p^{48} T^{18} +$$$$96\!\cdots\!49$$$$p^{56} T^{19} +$$$$97\!\cdots\!62$$$$p^{64} T^{20} +$$$$20\!\cdots\!73$$$$p^{72} T^{21} + 235041725822940 p^{80} T^{22} + 36824553 p^{88} T^{23} + p^{96} T^{24}$$
71 $$( 1 - 15005778 T + 1741360781651802 T^{2} -$$$$18\!\cdots\!26$$$$T^{3} +$$$$15\!\cdots\!35$$$$T^{4} -$$$$67\!\cdots\!92$$$$T^{5} +$$$$99\!\cdots\!28$$$$T^{6} -$$$$67\!\cdots\!92$$$$p^{8} T^{7} +$$$$15\!\cdots\!35$$$$p^{16} T^{8} -$$$$18\!\cdots\!26$$$$p^{24} T^{9} + 1741360781651802 p^{32} T^{10} - 15005778 p^{40} T^{11} + p^{48} T^{12} )^{2}$$
73 $$1 - 95080185 T + 7788143704773942 T^{2} -$$$$45\!\cdots\!95$$$$T^{3} +$$$$23\!\cdots\!24$$$$T^{4} -$$$$10\!\cdots\!97$$$$T^{5} +$$$$44\!\cdots\!82$$$$T^{6} -$$$$16\!\cdots\!31$$$$T^{7} +$$$$60\!\cdots\!68$$$$T^{8} -$$$$20\!\cdots\!37$$$$T^{9} +$$$$66\!\cdots\!10$$$$T^{10} -$$$$20\!\cdots\!11$$$$T^{11} +$$$$58\!\cdots\!98$$$$T^{12} -$$$$20\!\cdots\!11$$$$p^{8} T^{13} +$$$$66\!\cdots\!10$$$$p^{16} T^{14} -$$$$20\!\cdots\!37$$$$p^{24} T^{15} +$$$$60\!\cdots\!68$$$$p^{32} T^{16} -$$$$16\!\cdots\!31$$$$p^{40} T^{17} +$$$$44\!\cdots\!82$$$$p^{48} T^{18} -$$$$10\!\cdots\!97$$$$p^{56} T^{19} +$$$$23\!\cdots\!24$$$$p^{64} T^{20} -$$$$45\!\cdots\!95$$$$p^{72} T^{21} + 7788143704773942 p^{80} T^{22} - 95080185 p^{88} T^{23} + p^{96} T^{24}$$
79 $$1 - 8514456 T - 5984367856004523 T^{2} +$$$$11\!\cdots\!72$$$$p T^{3} +$$$$17\!\cdots\!63$$$$T^{4} -$$$$37\!\cdots\!36$$$$p T^{5} -$$$$38\!\cdots\!80$$$$T^{6} +$$$$40\!\cdots\!48$$$$T^{7} +$$$$75\!\cdots\!69$$$$T^{8} -$$$$16\!\cdots\!28$$$$T^{9} -$$$$14\!\cdots\!53$$$$T^{10} -$$$$47\!\cdots\!80$$$$T^{11} +$$$$25\!\cdots\!46$$$$T^{12} -$$$$47\!\cdots\!80$$$$p^{8} T^{13} -$$$$14\!\cdots\!53$$$$p^{16} T^{14} -$$$$16\!\cdots\!28$$$$p^{24} T^{15} +$$$$75\!\cdots\!69$$$$p^{32} T^{16} +$$$$40\!\cdots\!48$$$$p^{40} T^{17} -$$$$38\!\cdots\!80$$$$p^{48} T^{18} -$$$$37\!\cdots\!36$$$$p^{57} T^{19} +$$$$17\!\cdots\!63$$$$p^{64} T^{20} +$$$$11\!\cdots\!72$$$$p^{73} T^{21} - 5984367856004523 p^{80} T^{22} - 8514456 p^{88} T^{23} + p^{96} T^{24}$$
83 $$1 - 9924703546116687 T^{2} +$$$$57\!\cdots\!79$$$$T^{4} -$$$$26\!\cdots\!48$$$$T^{6} +$$$$92\!\cdots\!29$$$$T^{8} -$$$$26\!\cdots\!13$$$$T^{10} +$$$$66\!\cdots\!42$$$$T^{12} -$$$$26\!\cdots\!13$$$$p^{16} T^{14} +$$$$92\!\cdots\!29$$$$p^{32} T^{16} -$$$$26\!\cdots\!48$$$$p^{48} T^{18} +$$$$57\!\cdots\!79$$$$p^{64} T^{20} - 9924703546116687 p^{80} T^{22} + p^{96} T^{24}$$
89 $$1 + 83038554 T + 15954309489433614 T^{2} +$$$$11\!\cdots\!68$$$$T^{3} +$$$$11\!\cdots\!17$$$$T^{4} +$$$$65\!\cdots\!84$$$$T^{5} +$$$$51\!\cdots\!46$$$$T^{6} +$$$$20\!\cdots\!10$$$$T^{7} +$$$$16\!\cdots\!82$$$$T^{8} +$$$$24\!\cdots\!74$$$$T^{9} +$$$$29\!\cdots\!54$$$$T^{10} -$$$$77\!\cdots\!52$$$$T^{11} +$$$$38\!\cdots\!37$$$$T^{12} -$$$$77\!\cdots\!52$$$$p^{8} T^{13} +$$$$29\!\cdots\!54$$$$p^{16} T^{14} +$$$$24\!\cdots\!74$$$$p^{24} T^{15} +$$$$16\!\cdots\!82$$$$p^{32} T^{16} +$$$$20\!\cdots\!10$$$$p^{40} T^{17} +$$$$51\!\cdots\!46$$$$p^{48} T^{18} +$$$$65\!\cdots\!84$$$$p^{56} T^{19} +$$$$11\!\cdots\!17$$$$p^{64} T^{20} +$$$$11\!\cdots\!68$$$$p^{72} T^{21} + 15954309489433614 p^{80} T^{22} + 83038554 p^{88} T^{23} + p^{96} T^{24}$$
97 $$1 - 76055551527339963 T^{2} +$$$$27\!\cdots\!15$$$$T^{4} -$$$$60\!\cdots\!12$$$$T^{6} +$$$$93\!\cdots\!37$$$$T^{8} -$$$$10\!\cdots\!61$$$$T^{10} +$$$$96\!\cdots\!06$$$$T^{12} -$$$$10\!\cdots\!61$$$$p^{16} T^{14} +$$$$93\!\cdots\!37$$$$p^{32} T^{16} -$$$$60\!\cdots\!12$$$$p^{48} T^{18} +$$$$27\!\cdots\!15$$$$p^{64} T^{20} - 76055551527339963 p^{80} T^{22} + p^{96} T^{24}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$