# Properties

 Degree $24$ Conductor $9.066\times 10^{18}$ Sign $1$ Motivic weight $3$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 9·3-s + 21·7-s − 16·8-s + 27·9-s − 9·11-s + 39·13-s + 69·17-s − 462·19-s − 189·21-s − 66·23-s + 144·24-s + 171·25-s − 18·27-s + 159·29-s + 72·31-s + 81·33-s + 1.11e3·37-s − 351·39-s + 147·41-s − 117·43-s + 783·47-s + 1.95e3·49-s − 621·51-s − 249·53-s − 336·56-s + 4.15e3·57-s − 4.24e3·59-s + ⋯
 L(s)  = 1 − 1.73·3-s + 1.13·7-s − 0.707·8-s + 9-s − 0.246·11-s + 0.832·13-s + 0.984·17-s − 5.57·19-s − 1.96·21-s − 0.598·23-s + 1.22·24-s + 1.36·25-s − 0.128·27-s + 1.01·29-s + 0.417·31-s + 0.427·33-s + 4.95·37-s − 1.44·39-s + 0.559·41-s − 0.414·43-s + 2.43·47-s + 5.70·49-s − 1.70·51-s − 0.645·53-s − 0.801·56-s + 9.66·57-s − 9.37·59-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$24$$ Conductor: $$2^{12} \cdot 19^{12}$$ Sign: $1$ Motivic weight: $$3$$ Character: induced by $\chi_{38} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(24,\ 2^{12} \cdot 19^{12} ,\ ( \ : [3/2]^{12} ),\ 1 )$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.66097$$ $$L(\frac12)$$ $$\approx$$ $$1.66097$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$( 1 + p^{3} T^{3} + p^{6} T^{6} )^{2}$$
19 $$1 + 462 T + 119769 T^{2} + 1138067 p T^{3} + 8295507 p^{2} T^{4} + 48456009 p^{3} T^{5} + 232420374 p^{4} T^{6} + 48456009 p^{6} T^{7} + 8295507 p^{8} T^{8} + 1138067 p^{10} T^{9} + 119769 p^{12} T^{10} + 462 p^{15} T^{11} + p^{18} T^{12}$$
good3 $$1 + p^{2} T + 2 p^{3} T^{2} + 29 p^{2} T^{3} + 7 p^{4} T^{4} - 44 p^{2} T^{5} - 4724 T^{6} + 647 p^{2} T^{7} + 18802 p^{3} T^{8} + 267433 p^{2} T^{9} - 16606 p^{4} T^{10} - 8562682 p^{2} T^{11} - 485342900 T^{12} - 8562682 p^{5} T^{13} - 16606 p^{10} T^{14} + 267433 p^{11} T^{15} + 18802 p^{15} T^{16} + 647 p^{17} T^{17} - 4724 p^{18} T^{18} - 44 p^{23} T^{19} + 7 p^{28} T^{20} + 29 p^{29} T^{21} + 2 p^{33} T^{22} + p^{35} T^{23} + p^{36} T^{24}$$
5 $$1 - 171 T^{2} - 432 T^{3} + 684 T^{4} + 189297 T^{5} + 2035721 T^{6} - 20411757 T^{7} - 39405582 T^{8} - 505727712 p T^{9} - 942467184 p T^{10} + 2343884688 p^{3} T^{11} - 629941358634 T^{12} + 2343884688 p^{6} T^{13} - 942467184 p^{7} T^{14} - 505727712 p^{10} T^{15} - 39405582 p^{12} T^{16} - 20411757 p^{15} T^{17} + 2035721 p^{18} T^{18} + 189297 p^{21} T^{19} + 684 p^{24} T^{20} - 432 p^{27} T^{21} - 171 p^{30} T^{22} + p^{36} T^{24}$$
7 $$1 - 3 p T - 1515 T^{2} + 3540 p T^{3} + 1512579 T^{4} - 17880063 T^{5} - 1075565731 T^{6} + 8262030114 T^{7} + 606007378278 T^{8} - 2582097624714 T^{9} - 275006022177042 T^{10} + 351692577138024 T^{11} + 103612362769606380 T^{12} + 351692577138024 p^{3} T^{13} - 275006022177042 p^{6} T^{14} - 2582097624714 p^{9} T^{15} + 606007378278 p^{12} T^{16} + 8262030114 p^{15} T^{17} - 1075565731 p^{18} T^{18} - 17880063 p^{21} T^{19} + 1512579 p^{24} T^{20} + 3540 p^{28} T^{21} - 1515 p^{30} T^{22} - 3 p^{34} T^{23} + p^{36} T^{24}$$
11 $$1 + 9 T - 3519 T^{2} + 24496 T^{3} + 5114331 T^{4} - 110124705 T^{5} - 2271860931 T^{6} + 52499450874 T^{7} + 731386484418 T^{8} + 207758908815234 T^{9} - 13017969765518466 T^{10} - 204189988973228292 T^{11} + 30443947856053596940 T^{12} - 204189988973228292 p^{3} T^{13} - 13017969765518466 p^{6} T^{14} + 207758908815234 p^{9} T^{15} + 731386484418 p^{12} T^{16} + 52499450874 p^{15} T^{17} - 2271860931 p^{18} T^{18} - 110124705 p^{21} T^{19} + 5114331 p^{24} T^{20} + 24496 p^{27} T^{21} - 3519 p^{30} T^{22} + 9 p^{33} T^{23} + p^{36} T^{24}$$
13 $$1 - 3 p T + 3093 T^{2} - 98440 T^{3} + 11349087 T^{4} - 378701979 T^{5} + 22862720507 T^{6} - 1116317647566 T^{7} + 56385479661678 T^{8} - 3687796202409564 T^{9} + 159033549430554174 T^{10} - 10099287187718310156 T^{11} +$$$$30\!\cdots\!88$$$$T^{12} - 10099287187718310156 p^{3} T^{13} + 159033549430554174 p^{6} T^{14} - 3687796202409564 p^{9} T^{15} + 56385479661678 p^{12} T^{16} - 1116317647566 p^{15} T^{17} + 22862720507 p^{18} T^{18} - 378701979 p^{21} T^{19} + 11349087 p^{24} T^{20} - 98440 p^{27} T^{21} + 3093 p^{30} T^{22} - 3 p^{34} T^{23} + p^{36} T^{24}$$
17 $$1 - 69 T + 6573 T^{2} + 298204 T^{3} - 55154319 T^{4} + 270492363 p T^{5} - 148240032933 T^{6} - 16787667391872 T^{7} + 1892272804048104 T^{8} - 125048706972526476 T^{9} + 913900858267910370 T^{10} + 33791091237853613706 p T^{11} -$$$$17\!\cdots\!28$$$$p^{2} T^{12} + 33791091237853613706 p^{4} T^{13} + 913900858267910370 p^{6} T^{14} - 125048706972526476 p^{9} T^{15} + 1892272804048104 p^{12} T^{16} - 16787667391872 p^{15} T^{17} - 148240032933 p^{18} T^{18} + 270492363 p^{22} T^{19} - 55154319 p^{24} T^{20} + 298204 p^{27} T^{21} + 6573 p^{30} T^{22} - 69 p^{33} T^{23} + p^{36} T^{24}$$
23 $$1 + 66 T - 9393 T^{2} + 671695 T^{3} + 134377509 T^{4} - 14874284241 T^{5} + 71997450171 T^{6} + 263342907791031 T^{7} - 15774109169407374 T^{8} - 4876569355971214692 T^{9} +$$$$29\!\cdots\!14$$$$T^{10} +$$$$83\!\cdots\!47$$$$T^{11} -$$$$84\!\cdots\!52$$$$T^{12} +$$$$83\!\cdots\!47$$$$p^{3} T^{13} +$$$$29\!\cdots\!14$$$$p^{6} T^{14} - 4876569355971214692 p^{9} T^{15} - 15774109169407374 p^{12} T^{16} + 263342907791031 p^{15} T^{17} + 71997450171 p^{18} T^{18} - 14874284241 p^{21} T^{19} + 134377509 p^{24} T^{20} + 671695 p^{27} T^{21} - 9393 p^{30} T^{22} + 66 p^{33} T^{23} + p^{36} T^{24}$$
29 $$1 - 159 T + 36549 T^{2} - 3331656 T^{3} + 421841088 T^{4} - 3272692197 p T^{5} + 5535241134320 T^{6} + 625908025808775 T^{7} + 1040540252412762 p T^{8} - 48469236719521320171 T^{9} +$$$$14\!\cdots\!33$$$$T^{10} -$$$$23\!\cdots\!52$$$$T^{11} +$$$$17\!\cdots\!06$$$$T^{12} -$$$$23\!\cdots\!52$$$$p^{3} T^{13} +$$$$14\!\cdots\!33$$$$p^{6} T^{14} - 48469236719521320171 p^{9} T^{15} + 1040540252412762 p^{13} T^{16} + 625908025808775 p^{15} T^{17} + 5535241134320 p^{18} T^{18} - 3272692197 p^{22} T^{19} + 421841088 p^{24} T^{20} - 3331656 p^{27} T^{21} + 36549 p^{30} T^{22} - 159 p^{33} T^{23} + p^{36} T^{24}$$
31 $$1 - 72 T - 84360 T^{2} - 4099822 T^{3} + 4645966785 T^{4} + 582123802836 T^{5} - 97979947426741 T^{6} - 42244835246634435 T^{7} - 818163153401404386 T^{8} +$$$$13\!\cdots\!80$$$$T^{9} +$$$$66\!\cdots\!43$$$$p T^{10} -$$$$20\!\cdots\!07$$$$T^{11} -$$$$78\!\cdots\!40$$$$T^{12} -$$$$20\!\cdots\!07$$$$p^{3} T^{13} +$$$$66\!\cdots\!43$$$$p^{7} T^{14} +$$$$13\!\cdots\!80$$$$p^{9} T^{15} - 818163153401404386 p^{12} T^{16} - 42244835246634435 p^{15} T^{17} - 97979947426741 p^{18} T^{18} + 582123802836 p^{21} T^{19} + 4645966785 p^{24} T^{20} - 4099822 p^{27} T^{21} - 84360 p^{30} T^{22} - 72 p^{33} T^{23} + p^{36} T^{24}$$
37 $$( 1 - 558 T + 298089 T^{2} - 113545809 T^{3} + 38178310329 T^{4} - 10429273769733 T^{5} + 69667380457282 p T^{6} - 10429273769733 p^{3} T^{7} + 38178310329 p^{6} T^{8} - 113545809 p^{9} T^{9} + 298089 p^{12} T^{10} - 558 p^{15} T^{11} + p^{18} T^{12} )^{2}$$
41 $$1 - 147 T + 44007 T^{2} + 21769246 T^{3} - 2350025733 T^{4} + 234418847265 T^{5} + 457706944154289 T^{6} - 9303997209190392 T^{7} - 6140319093154185000 T^{8} +$$$$10\!\cdots\!34$$$$T^{9} -$$$$12\!\cdots\!74$$$$T^{10} -$$$$45\!\cdots\!46$$$$T^{11} +$$$$16\!\cdots\!04$$$$T^{12} -$$$$45\!\cdots\!46$$$$p^{3} T^{13} -$$$$12\!\cdots\!74$$$$p^{6} T^{14} +$$$$10\!\cdots\!34$$$$p^{9} T^{15} - 6140319093154185000 p^{12} T^{16} - 9303997209190392 p^{15} T^{17} + 457706944154289 p^{18} T^{18} + 234418847265 p^{21} T^{19} - 2350025733 p^{24} T^{20} + 21769246 p^{27} T^{21} + 44007 p^{30} T^{22} - 147 p^{33} T^{23} + p^{36} T^{24}$$
43 $$1 + 117 T - 100119 T^{2} + 38915762 T^{3} + 2994100635 T^{4} - 3671912050095 T^{5} + 1310121343946135 T^{6} - 70375225814813832 T^{7} - 74777917746500954580 T^{8} +$$$$41\!\cdots\!76$$$$T^{9} -$$$$21\!\cdots\!80$$$$T^{10} -$$$$11\!\cdots\!68$$$$T^{11} +$$$$88\!\cdots\!32$$$$T^{12} -$$$$11\!\cdots\!68$$$$p^{3} T^{13} -$$$$21\!\cdots\!80$$$$p^{6} T^{14} +$$$$41\!\cdots\!76$$$$p^{9} T^{15} - 74777917746500954580 p^{12} T^{16} - 70375225814813832 p^{15} T^{17} + 1310121343946135 p^{18} T^{18} - 3671912050095 p^{21} T^{19} + 2994100635 p^{24} T^{20} + 38915762 p^{27} T^{21} - 100119 p^{30} T^{22} + 117 p^{33} T^{23} + p^{36} T^{24}$$
47 $$1 - 783 T + 538857 T^{2} - 254728868 T^{3} + 112205777895 T^{4} - 37288296975249 T^{5} + 10875425952712257 T^{6} - 1895616375918277806 T^{7} + 46254528927632446254 T^{8} +$$$$26\!\cdots\!66$$$$T^{9} -$$$$31\!\cdots\!62$$$$p T^{10} +$$$$68\!\cdots\!52$$$$T^{11} -$$$$22\!\cdots\!72$$$$T^{12} +$$$$68\!\cdots\!52$$$$p^{3} T^{13} -$$$$31\!\cdots\!62$$$$p^{7} T^{14} +$$$$26\!\cdots\!66$$$$p^{9} T^{15} + 46254528927632446254 p^{12} T^{16} - 1895616375918277806 p^{15} T^{17} + 10875425952712257 p^{18} T^{18} - 37288296975249 p^{21} T^{19} + 112205777895 p^{24} T^{20} - 254728868 p^{27} T^{21} + 538857 p^{30} T^{22} - 783 p^{33} T^{23} + p^{36} T^{24}$$
53 $$1 + 249 T + 105357 T^{2} + 53055920 T^{3} + 56278965819 T^{4} + 10000173124845 T^{5} + 3156327827474091 T^{6} + 2244063380209899702 T^{7} +$$$$94\!\cdots\!94$$$$T^{8} +$$$$24\!\cdots\!32$$$$T^{9} +$$$$31\!\cdots\!86$$$$T^{10} +$$$$47\!\cdots\!40$$$$T^{11} +$$$$98\!\cdots\!48$$$$T^{12} +$$$$47\!\cdots\!40$$$$p^{3} T^{13} +$$$$31\!\cdots\!86$$$$p^{6} T^{14} +$$$$24\!\cdots\!32$$$$p^{9} T^{15} +$$$$94\!\cdots\!94$$$$p^{12} T^{16} + 2244063380209899702 p^{15} T^{17} + 3156327827474091 p^{18} T^{18} + 10000173124845 p^{21} T^{19} + 56278965819 p^{24} T^{20} + 53055920 p^{27} T^{21} + 105357 p^{30} T^{22} + 249 p^{33} T^{23} + p^{36} T^{24}$$
59 $$1 + 72 p T + 147897 p T^{2} + 11515262701 T^{3} + 10887706714419 T^{4} + 7727213299360905 T^{5} + 4135284193453945767 T^{6} +$$$$15\!\cdots\!07$$$$T^{7} +$$$$29\!\cdots\!98$$$$T^{8} -$$$$11\!\cdots\!96$$$$T^{9} -$$$$15\!\cdots\!02$$$$T^{10} -$$$$10\!\cdots\!07$$$$T^{11} -$$$$50\!\cdots\!20$$$$T^{12} -$$$$10\!\cdots\!07$$$$p^{3} T^{13} -$$$$15\!\cdots\!02$$$$p^{6} T^{14} -$$$$11\!\cdots\!96$$$$p^{9} T^{15} +$$$$29\!\cdots\!98$$$$p^{12} T^{16} +$$$$15\!\cdots\!07$$$$p^{15} T^{17} + 4135284193453945767 p^{18} T^{18} + 7727213299360905 p^{21} T^{19} + 10887706714419 p^{24} T^{20} + 11515262701 p^{27} T^{21} + 147897 p^{31} T^{22} + 72 p^{34} T^{23} + p^{36} T^{24}$$
61 $$1 - 3114 T + 4298859 T^{2} - 3204919049 T^{3} + 998006352417 T^{4} + 482660655435063 T^{5} - 719435635956944909 T^{6} +$$$$34\!\cdots\!33$$$$T^{7} -$$$$35\!\cdots\!32$$$$T^{8} -$$$$50\!\cdots\!02$$$$T^{9} +$$$$27\!\cdots\!08$$$$T^{10} -$$$$17\!\cdots\!79$$$$T^{11} -$$$$25\!\cdots\!48$$$$T^{12} -$$$$17\!\cdots\!79$$$$p^{3} T^{13} +$$$$27\!\cdots\!08$$$$p^{6} T^{14} -$$$$50\!\cdots\!02$$$$p^{9} T^{15} -$$$$35\!\cdots\!32$$$$p^{12} T^{16} +$$$$34\!\cdots\!33$$$$p^{15} T^{17} - 719435635956944909 p^{18} T^{18} + 482660655435063 p^{21} T^{19} + 998006352417 p^{24} T^{20} - 3204919049 p^{27} T^{21} + 4298859 p^{30} T^{22} - 3114 p^{33} T^{23} + p^{36} T^{24}$$
67 $$1 - 3060 T + 4343541 T^{2} - 3539177707 T^{3} + 1674046544745 T^{4} - 364838652101685 T^{5} - 33128397443652253 T^{6} + 59458849334550877089 T^{7} -$$$$73\!\cdots\!64$$$$T^{8} +$$$$81\!\cdots\!52$$$$T^{9} -$$$$50\!\cdots\!24$$$$T^{10} +$$$$16\!\cdots\!97$$$$T^{11} -$$$$43\!\cdots\!88$$$$T^{12} +$$$$16\!\cdots\!97$$$$p^{3} T^{13} -$$$$50\!\cdots\!24$$$$p^{6} T^{14} +$$$$81\!\cdots\!52$$$$p^{9} T^{15} -$$$$73\!\cdots\!64$$$$p^{12} T^{16} + 59458849334550877089 p^{15} T^{17} - 33128397443652253 p^{18} T^{18} - 364838652101685 p^{21} T^{19} + 1674046544745 p^{24} T^{20} - 3539177707 p^{27} T^{21} + 4343541 p^{30} T^{22} - 3060 p^{33} T^{23} + p^{36} T^{24}$$
71 $$1 - 1686 T + 18357 p T^{2} - 332046979 T^{3} - 395700826839 T^{4} + 496154621895921 T^{5} - 192091304753971689 T^{6} -$$$$10\!\cdots\!95$$$$T^{7} +$$$$20\!\cdots\!46$$$$T^{8} -$$$$13\!\cdots\!76$$$$T^{9} +$$$$27\!\cdots\!34$$$$T^{10} +$$$$26\!\cdots\!85$$$$T^{11} -$$$$28\!\cdots\!24$$$$T^{12} +$$$$26\!\cdots\!85$$$$p^{3} T^{13} +$$$$27\!\cdots\!34$$$$p^{6} T^{14} -$$$$13\!\cdots\!76$$$$p^{9} T^{15} +$$$$20\!\cdots\!46$$$$p^{12} T^{16} -$$$$10\!\cdots\!95$$$$p^{15} T^{17} - 192091304753971689 p^{18} T^{18} + 496154621895921 p^{21} T^{19} - 395700826839 p^{24} T^{20} - 332046979 p^{27} T^{21} + 18357 p^{31} T^{22} - 1686 p^{33} T^{23} + p^{36} T^{24}$$
73 $$1 - 1626 T + 1342599 T^{2} - 77826027 T^{3} - 482426009259 T^{4} + 337597036991529 T^{5} + 152387349969643313 T^{6} -$$$$21\!\cdots\!53$$$$T^{7} +$$$$80\!\cdots\!70$$$$T^{8} +$$$$61\!\cdots\!48$$$$T^{9} -$$$$37\!\cdots\!06$$$$T^{10} -$$$$65\!\cdots\!35$$$$T^{11} +$$$$17\!\cdots\!44$$$$T^{12} -$$$$65\!\cdots\!35$$$$p^{3} T^{13} -$$$$37\!\cdots\!06$$$$p^{6} T^{14} +$$$$61\!\cdots\!48$$$$p^{9} T^{15} +$$$$80\!\cdots\!70$$$$p^{12} T^{16} -$$$$21\!\cdots\!53$$$$p^{15} T^{17} + 152387349969643313 p^{18} T^{18} + 337597036991529 p^{21} T^{19} - 482426009259 p^{24} T^{20} - 77826027 p^{27} T^{21} + 1342599 p^{30} T^{22} - 1626 p^{33} T^{23} + p^{36} T^{24}$$
79 $$1 + 327 T - 352623 T^{2} + 85160886 T^{3} - 74868578124 T^{4} - 561682606508301 T^{5} - 10832178836590636 T^{6} +$$$$15\!\cdots\!35$$$$T^{7} -$$$$13\!\cdots\!24$$$$T^{8} +$$$$11\!\cdots\!49$$$$T^{9} +$$$$69\!\cdots\!71$$$$T^{10} -$$$$11\!\cdots\!46$$$$T^{11} -$$$$30\!\cdots\!66$$$$T^{12} -$$$$11\!\cdots\!46$$$$p^{3} T^{13} +$$$$69\!\cdots\!71$$$$p^{6} T^{14} +$$$$11\!\cdots\!49$$$$p^{9} T^{15} -$$$$13\!\cdots\!24$$$$p^{12} T^{16} +$$$$15\!\cdots\!35$$$$p^{15} T^{17} - 10832178836590636 p^{18} T^{18} - 561682606508301 p^{21} T^{19} - 74868578124 p^{24} T^{20} + 85160886 p^{27} T^{21} - 352623 p^{30} T^{22} + 327 p^{33} T^{23} + p^{36} T^{24}$$
83 $$1 - 927 T - 1910169 T^{2} + 1162039358 T^{3} + 2637336171501 T^{4} - 771225603730341 T^{5} - 2584240489804536525 T^{6} +$$$$27\!\cdots\!66$$$$T^{7} +$$$$19\!\cdots\!86$$$$T^{8} -$$$$17\!\cdots\!40$$$$T^{9} -$$$$12\!\cdots\!02$$$$T^{10} -$$$$85\!\cdots\!76$$$$T^{11} +$$$$72\!\cdots\!04$$$$T^{12} -$$$$85\!\cdots\!76$$$$p^{3} T^{13} -$$$$12\!\cdots\!02$$$$p^{6} T^{14} -$$$$17\!\cdots\!40$$$$p^{9} T^{15} +$$$$19\!\cdots\!86$$$$p^{12} T^{16} +$$$$27\!\cdots\!66$$$$p^{15} T^{17} - 2584240489804536525 p^{18} T^{18} - 771225603730341 p^{21} T^{19} + 2637336171501 p^{24} T^{20} + 1162039358 p^{27} T^{21} - 1910169 p^{30} T^{22} - 927 p^{33} T^{23} + p^{36} T^{24}$$
89 $$1 + 6366 T + 18939837 T^{2} + 33490901356 T^{3} + 36794607216366 T^{4} + 23062320055435131 T^{5} + 6331029765277091109 T^{6} +$$$$65\!\cdots\!65$$$$T^{7} +$$$$26\!\cdots\!56$$$$p T^{8} +$$$$34\!\cdots\!44$$$$T^{9} +$$$$26\!\cdots\!92$$$$T^{10} +$$$$82\!\cdots\!30$$$$T^{11} +$$$$98\!\cdots\!30$$$$T^{12} +$$$$82\!\cdots\!30$$$$p^{3} T^{13} +$$$$26\!\cdots\!92$$$$p^{6} T^{14} +$$$$34\!\cdots\!44$$$$p^{9} T^{15} +$$$$26\!\cdots\!56$$$$p^{13} T^{16} +$$$$65\!\cdots\!65$$$$p^{15} T^{17} + 6331029765277091109 p^{18} T^{18} + 23062320055435131 p^{21} T^{19} + 36794607216366 p^{24} T^{20} + 33490901356 p^{27} T^{21} + 18939837 p^{30} T^{22} + 6366 p^{33} T^{23} + p^{36} T^{24}$$
97 $$1 + 8052 T + 31462761 T^{2} + 80299670389 T^{3} + 153146011957221 T^{4} + 239405956107270819 T^{5} +$$$$33\!\cdots\!19$$$$T^{6} +$$$$42\!\cdots\!83$$$$T^{7} +$$$$52\!\cdots\!34$$$$T^{8} +$$$$60\!\cdots\!74$$$$T^{9} +$$$$65\!\cdots\!46$$$$T^{10} +$$$$66\!\cdots\!83$$$$T^{11} +$$$$64\!\cdots\!84$$$$T^{12} +$$$$66\!\cdots\!83$$$$p^{3} T^{13} +$$$$65\!\cdots\!46$$$$p^{6} T^{14} +$$$$60\!\cdots\!74$$$$p^{9} T^{15} +$$$$52\!\cdots\!34$$$$p^{12} T^{16} +$$$$42\!\cdots\!83$$$$p^{15} T^{17} +$$$$33\!\cdots\!19$$$$p^{18} T^{18} + 239405956107270819 p^{21} T^{19} + 153146011957221 p^{24} T^{20} + 80299670389 p^{27} T^{21} + 31462761 p^{30} T^{22} + 8052 p^{33} T^{23} + p^{36} T^{24}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$