# Properties

 Label 24-17e24-1.1-c3e12-0-2 Degree $24$ Conductor $3.394\times 10^{29}$ Sign $1$ Analytic cond. $6.04180\times 10^{14}$ Root an. cond. $4.12935$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $12$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 18·3-s − 24·4-s − 30·5-s − 24·7-s − 17·8-s + 54·9-s − 162·11-s + 432·12-s − 72·13-s + 540·15-s + 240·16-s − 66·19-s + 720·20-s + 432·21-s − 282·23-s + 306·24-s − 78·25-s + 770·27-s + 576·28-s − 648·29-s − 504·31-s + 219·32-s + 2.91e3·33-s + 720·35-s − 1.29e3·36-s + 30·37-s + 1.29e3·39-s + ⋯
 L(s)  = 1 − 3.46·3-s − 3·4-s − 2.68·5-s − 1.29·7-s − 0.751·8-s + 2·9-s − 4.44·11-s + 10.3·12-s − 1.53·13-s + 9.29·15-s + 15/4·16-s − 0.796·19-s + 8.04·20-s + 4.48·21-s − 2.55·23-s + 2.60·24-s − 0.623·25-s + 5.48·27-s + 3.88·28-s − 4.14·29-s − 2.92·31-s + 1.20·32-s + 15.3·33-s + 3.47·35-s − 6·36-s + 0.133·37-s + 5.32·39-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{24}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$24$$ Conductor: $$17^{24}$$ Sign: $1$ Analytic conductor: $$6.04180\times 10^{14}$$ Root analytic conductor: $$4.12935$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$12$$ Selberg data: $$(24,\ 17^{24} ,\ ( \ : [3/2]^{12} ),\ 1 )$$

## Particular Values

 $$L(2)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad17 $$1$$
good2 $$1 + 3 p^{3} T^{2} + 17 T^{3} + 21 p^{4} T^{4} + 597 T^{5} + 2083 p T^{6} + 9219 T^{7} + 47463 T^{8} + 99575 T^{9} + 237969 p T^{10} + 29085 p^{5} T^{11} + 516393 p^{3} T^{12} + 29085 p^{8} T^{13} + 237969 p^{7} T^{14} + 99575 p^{9} T^{15} + 47463 p^{12} T^{16} + 9219 p^{15} T^{17} + 2083 p^{19} T^{18} + 597 p^{21} T^{19} + 21 p^{28} T^{20} + 17 p^{27} T^{21} + 3 p^{33} T^{22} + p^{36} T^{24}$$
3 $$1 + 2 p^{2} T + 10 p^{3} T^{2} + 3118 T^{3} + 3493 p^{2} T^{4} + 93338 p T^{5} + 2283230 T^{6} + 5663386 p T^{7} + 39342911 p T^{8} + 761340866 T^{9} + 1539722494 p T^{10} + 972382312 p^{3} T^{11} + 46929762953 p T^{12} + 972382312 p^{6} T^{13} + 1539722494 p^{7} T^{14} + 761340866 p^{9} T^{15} + 39342911 p^{13} T^{16} + 5663386 p^{16} T^{17} + 2283230 p^{18} T^{18} + 93338 p^{22} T^{19} + 3493 p^{26} T^{20} + 3118 p^{27} T^{21} + 10 p^{33} T^{22} + 2 p^{35} T^{23} + p^{36} T^{24}$$
5 $$1 + 6 p T + 978 T^{2} + 4402 p T^{3} + 98928 p T^{4} + 8963928 T^{5} + 160010892 T^{6} + 2486105382 T^{7} + 37688092122 T^{8} + 511113233924 T^{9} + 6768898744506 T^{10} + 81309529724886 T^{11} + 952406498268277 T^{12} + 81309529724886 p^{3} T^{13} + 6768898744506 p^{6} T^{14} + 511113233924 p^{9} T^{15} + 37688092122 p^{12} T^{16} + 2486105382 p^{15} T^{17} + 160010892 p^{18} T^{18} + 8963928 p^{21} T^{19} + 98928 p^{25} T^{20} + 4402 p^{28} T^{21} + 978 p^{30} T^{22} + 6 p^{34} T^{23} + p^{36} T^{24}$$
7 $$1 + 24 T + 2631 T^{2} + 49932 T^{3} + 3178503 T^{4} + 50026356 T^{5} + 351286134 p T^{6} + 33717298626 T^{7} + 1420494335676 T^{8} + 17645494346682 T^{9} + 658479284458026 T^{10} + 7488753262542012 T^{11} + 249672863392321825 T^{12} + 7488753262542012 p^{3} T^{13} + 658479284458026 p^{6} T^{14} + 17645494346682 p^{9} T^{15} + 1420494335676 p^{12} T^{16} + 33717298626 p^{15} T^{17} + 351286134 p^{19} T^{18} + 50026356 p^{21} T^{19} + 3178503 p^{24} T^{20} + 49932 p^{27} T^{21} + 2631 p^{30} T^{22} + 24 p^{33} T^{23} + p^{36} T^{24}$$
11 $$1 + 162 T + 20376 T^{2} + 1758520 T^{3} + 128665887 T^{4} + 7549392330 T^{5} + 386047940887 T^{6} + 16321486458276 T^{7} + 600098184489765 T^{8} + 17707891221939426 T^{9} + 444740033046531609 T^{10} + 8519881241999961450 T^{11} +$$$$22\!\cdots\!34$$$$T^{12} + 8519881241999961450 p^{3} T^{13} + 444740033046531609 p^{6} T^{14} + 17707891221939426 p^{9} T^{15} + 600098184489765 p^{12} T^{16} + 16321486458276 p^{15} T^{17} + 386047940887 p^{18} T^{18} + 7549392330 p^{21} T^{19} + 128665887 p^{24} T^{20} + 1758520 p^{27} T^{21} + 20376 p^{30} T^{22} + 162 p^{33} T^{23} + p^{36} T^{24}$$
13 $$1 + 72 T + 14790 T^{2} + 952402 T^{3} + 109939374 T^{4} + 6632292606 T^{5} + 549660962912 T^{6} + 31473741682326 T^{7} + 2066167770898980 T^{8} + 111808560575162252 T^{9} + 6158767151543542860 T^{10} +$$$$30\!\cdots\!90$$$$T^{11} +$$$$14\!\cdots\!37$$$$T^{12} +$$$$30\!\cdots\!90$$$$p^{3} T^{13} + 6158767151543542860 p^{6} T^{14} + 111808560575162252 p^{9} T^{15} + 2066167770898980 p^{12} T^{16} + 31473741682326 p^{15} T^{17} + 549660962912 p^{18} T^{18} + 6632292606 p^{21} T^{19} + 109939374 p^{24} T^{20} + 952402 p^{27} T^{21} + 14790 p^{30} T^{22} + 72 p^{33} T^{23} + p^{36} T^{24}$$
19 $$1 + 66 T + 31476 T^{2} + 2467196 T^{3} + 644635554 T^{4} + 47808045678 T^{5} + 9459081995078 T^{6} + 672750203320836 T^{7} + 107985364016568894 T^{8} + 7139368709734578288 T^{9} +$$$$99\!\cdots\!78$$$$T^{10} +$$$$60\!\cdots\!64$$$$T^{11} +$$$$75\!\cdots\!51$$$$T^{12} +$$$$60\!\cdots\!64$$$$p^{3} T^{13} +$$$$99\!\cdots\!78$$$$p^{6} T^{14} + 7139368709734578288 p^{9} T^{15} + 107985364016568894 p^{12} T^{16} + 672750203320836 p^{15} T^{17} + 9459081995078 p^{18} T^{18} + 47808045678 p^{21} T^{19} + 644635554 p^{24} T^{20} + 2467196 p^{27} T^{21} + 31476 p^{30} T^{22} + 66 p^{33} T^{23} + p^{36} T^{24}$$
23 $$1 + 282 T + 129591 T^{2} + 26870378 T^{3} + 7295872581 T^{4} + 1217919573750 T^{5} + 248562967071774 T^{6} + 35006925558952032 T^{7} + 5876611919308179474 T^{8} +$$$$71\!\cdots\!46$$$$T^{9} +$$$$10\!\cdots\!10$$$$T^{10} +$$$$11\!\cdots\!86$$$$T^{11} +$$$$14\!\cdots\!77$$$$T^{12} +$$$$11\!\cdots\!86$$$$p^{3} T^{13} +$$$$10\!\cdots\!10$$$$p^{6} T^{14} +$$$$71\!\cdots\!46$$$$p^{9} T^{15} + 5876611919308179474 p^{12} T^{16} + 35006925558952032 p^{15} T^{17} + 248562967071774 p^{18} T^{18} + 1217919573750 p^{21} T^{19} + 7295872581 p^{24} T^{20} + 26870378 p^{27} T^{21} + 129591 p^{30} T^{22} + 282 p^{33} T^{23} + p^{36} T^{24}$$
29 $$1 + 648 T + 363351 T^{2} + 141812376 T^{3} + 49113689433 T^{4} + 14196586050282 T^{5} + 3739660471763232 T^{6} + 872788818202113846 T^{7} +$$$$18\!\cdots\!86$$$$T^{8} +$$$$37\!\cdots\!68$$$$T^{9} +$$$$69\!\cdots\!26$$$$T^{10} +$$$$11\!\cdots\!52$$$$T^{11} +$$$$19\!\cdots\!41$$$$T^{12} +$$$$11\!\cdots\!52$$$$p^{3} T^{13} +$$$$69\!\cdots\!26$$$$p^{6} T^{14} +$$$$37\!\cdots\!68$$$$p^{9} T^{15} +$$$$18\!\cdots\!86$$$$p^{12} T^{16} + 872788818202113846 p^{15} T^{17} + 3739660471763232 p^{18} T^{18} + 14196586050282 p^{21} T^{19} + 49113689433 p^{24} T^{20} + 141812376 p^{27} T^{21} + 363351 p^{30} T^{22} + 648 p^{33} T^{23} + p^{36} T^{24}$$
31 $$1 + 504 T + 287307 T^{2} + 103850426 T^{3} + 36701706309 T^{4} + 10521372364698 T^{5} + 2904783892693358 T^{6} + 703577060018454174 T^{7} +$$$$16\!\cdots\!48$$$$T^{8} +$$$$34\!\cdots\!54$$$$T^{9} +$$$$70\!\cdots\!84$$$$T^{10} +$$$$13\!\cdots\!86$$$$T^{11} +$$$$23\!\cdots\!89$$$$T^{12} +$$$$13\!\cdots\!86$$$$p^{3} T^{13} +$$$$70\!\cdots\!84$$$$p^{6} T^{14} +$$$$34\!\cdots\!54$$$$p^{9} T^{15} +$$$$16\!\cdots\!48$$$$p^{12} T^{16} + 703577060018454174 p^{15} T^{17} + 2904783892693358 p^{18} T^{18} + 10521372364698 p^{21} T^{19} + 36701706309 p^{24} T^{20} + 103850426 p^{27} T^{21} + 287307 p^{30} T^{22} + 504 p^{33} T^{23} + p^{36} T^{24}$$
37 $$1 - 30 T + 396441 T^{2} + 4303652 T^{3} + 75535258077 T^{4} + 3306733572444 T^{5} + 9386012826459812 T^{6} + 609429333975224280 T^{7} +$$$$86\!\cdots\!90$$$$T^{8} +$$$$63\!\cdots\!52$$$$T^{9} +$$$$61\!\cdots\!80$$$$T^{10} +$$$$43\!\cdots\!68$$$$T^{11} +$$$$35\!\cdots\!25$$$$T^{12} +$$$$43\!\cdots\!68$$$$p^{3} T^{13} +$$$$61\!\cdots\!80$$$$p^{6} T^{14} +$$$$63\!\cdots\!52$$$$p^{9} T^{15} +$$$$86\!\cdots\!90$$$$p^{12} T^{16} + 609429333975224280 p^{15} T^{17} + 9386012826459812 p^{18} T^{18} + 3306733572444 p^{21} T^{19} + 75535258077 p^{24} T^{20} + 4303652 p^{27} T^{21} + 396441 p^{30} T^{22} - 30 p^{33} T^{23} + p^{36} T^{24}$$
41 $$1 + 318 T + 461895 T^{2} + 140587038 T^{3} + 110124298701 T^{4} + 31262577390408 T^{5} + 431137996827126 p T^{6} + 4635187016534166306 T^{7} +$$$$21\!\cdots\!94$$$$T^{8} +$$$$51\!\cdots\!64$$$$T^{9} +$$$$20\!\cdots\!18$$$$T^{10} +$$$$44\!\cdots\!00$$$$T^{11} +$$$$15\!\cdots\!79$$$$T^{12} +$$$$44\!\cdots\!00$$$$p^{3} T^{13} +$$$$20\!\cdots\!18$$$$p^{6} T^{14} +$$$$51\!\cdots\!64$$$$p^{9} T^{15} +$$$$21\!\cdots\!94$$$$p^{12} T^{16} + 4635187016534166306 p^{15} T^{17} + 431137996827126 p^{19} T^{18} + 31262577390408 p^{21} T^{19} + 110124298701 p^{24} T^{20} + 140587038 p^{27} T^{21} + 461895 p^{30} T^{22} + 318 p^{33} T^{23} + p^{36} T^{24}$$
43 $$1 - 486 T + 537816 T^{2} - 234810144 T^{3} + 149332540095 T^{4} - 58372416122220 T^{5} + 28190059917868464 T^{6} - 9841117987706330736 T^{7} +$$$$39\!\cdots\!47$$$$T^{8} -$$$$12\!\cdots\!82$$$$T^{9} +$$$$43\!\cdots\!10$$$$T^{10} -$$$$28\!\cdots\!64$$$$p T^{11} +$$$$38\!\cdots\!75$$$$T^{12} -$$$$28\!\cdots\!64$$$$p^{4} T^{13} +$$$$43\!\cdots\!10$$$$p^{6} T^{14} -$$$$12\!\cdots\!82$$$$p^{9} T^{15} +$$$$39\!\cdots\!47$$$$p^{12} T^{16} - 9841117987706330736 p^{15} T^{17} + 28190059917868464 p^{18} T^{18} - 58372416122220 p^{21} T^{19} + 149332540095 p^{24} T^{20} - 234810144 p^{27} T^{21} + 537816 p^{30} T^{22} - 486 p^{33} T^{23} + p^{36} T^{24}$$
47 $$1 + 888 T + 926349 T^{2} + 650503016 T^{3} + 421950918861 T^{4} + 238133559715818 T^{5} + 122068250658909346 T^{6} + 57411272664754757682 T^{7} +$$$$24\!\cdots\!54$$$$T^{8} +$$$$10\!\cdots\!46$$$$T^{9} +$$$$38\!\cdots\!66$$$$T^{10} +$$$$13\!\cdots\!86$$$$T^{11} +$$$$44\!\cdots\!93$$$$T^{12} +$$$$13\!\cdots\!86$$$$p^{3} T^{13} +$$$$38\!\cdots\!66$$$$p^{6} T^{14} +$$$$10\!\cdots\!46$$$$p^{9} T^{15} +$$$$24\!\cdots\!54$$$$p^{12} T^{16} + 57411272664754757682 p^{15} T^{17} + 122068250658909346 p^{18} T^{18} + 238133559715818 p^{21} T^{19} + 421950918861 p^{24} T^{20} + 650503016 p^{27} T^{21} + 926349 p^{30} T^{22} + 888 p^{33} T^{23} + p^{36} T^{24}$$
53 $$1 - 1026 T + 1814271 T^{2} - 26381752 p T^{3} + 1427421860244 T^{4} - 890940789162870 T^{5} + 671398014573953230 T^{6} -$$$$35\!\cdots\!16$$$$T^{7} +$$$$21\!\cdots\!45$$$$T^{8} -$$$$96\!\cdots\!94$$$$T^{9} +$$$$49\!\cdots\!87$$$$T^{10} -$$$$19\!\cdots\!78$$$$T^{11} +$$$$85\!\cdots\!88$$$$T^{12} -$$$$19\!\cdots\!78$$$$p^{3} T^{13} +$$$$49\!\cdots\!87$$$$p^{6} T^{14} -$$$$96\!\cdots\!94$$$$p^{9} T^{15} +$$$$21\!\cdots\!45$$$$p^{12} T^{16} -$$$$35\!\cdots\!16$$$$p^{15} T^{17} + 671398014573953230 p^{18} T^{18} - 890940789162870 p^{21} T^{19} + 1427421860244 p^{24} T^{20} - 26381752 p^{28} T^{21} + 1814271 p^{30} T^{22} - 1026 p^{33} T^{23} + p^{36} T^{24}$$
59 $$1 + 792 T + 32886 p T^{2} + 1404683820 T^{3} + 1829013156759 T^{4} + 1178531621654760 T^{5} + 1092756627068295240 T^{6} +$$$$62\!\cdots\!28$$$$T^{7} +$$$$45\!\cdots\!01$$$$T^{8} +$$$$23\!\cdots\!76$$$$T^{9} +$$$$14\!\cdots\!54$$$$T^{10} +$$$$62\!\cdots\!40$$$$T^{11} +$$$$33\!\cdots\!35$$$$T^{12} +$$$$62\!\cdots\!40$$$$p^{3} T^{13} +$$$$14\!\cdots\!54$$$$p^{6} T^{14} +$$$$23\!\cdots\!76$$$$p^{9} T^{15} +$$$$45\!\cdots\!01$$$$p^{12} T^{16} +$$$$62\!\cdots\!28$$$$p^{15} T^{17} + 1092756627068295240 p^{18} T^{18} + 1178531621654760 p^{21} T^{19} + 1829013156759 p^{24} T^{20} + 1404683820 p^{27} T^{21} + 32886 p^{31} T^{22} + 792 p^{33} T^{23} + p^{36} T^{24}$$
61 $$1 + 1212 T + 2239122 T^{2} + 1950405742 T^{3} + 2103563986245 T^{4} + 1463801267573628 T^{5} + 1192783852845337544 T^{6} +$$$$70\!\cdots\!50$$$$T^{7} +$$$$48\!\cdots\!71$$$$T^{8} +$$$$25\!\cdots\!88$$$$T^{9} +$$$$15\!\cdots\!48$$$$T^{10} +$$$$71\!\cdots\!12$$$$T^{11} +$$$$38\!\cdots\!85$$$$T^{12} +$$$$71\!\cdots\!12$$$$p^{3} T^{13} +$$$$15\!\cdots\!48$$$$p^{6} T^{14} +$$$$25\!\cdots\!88$$$$p^{9} T^{15} +$$$$48\!\cdots\!71$$$$p^{12} T^{16} +$$$$70\!\cdots\!50$$$$p^{15} T^{17} + 1192783852845337544 p^{18} T^{18} + 1463801267573628 p^{21} T^{19} + 2103563986245 p^{24} T^{20} + 1950405742 p^{27} T^{21} + 2239122 p^{30} T^{22} + 1212 p^{33} T^{23} + p^{36} T^{24}$$
67 $$1 - 624 T + 2227497 T^{2} - 1020875822 T^{3} + 2240105622237 T^{4} - 734700728026800 T^{5} + 1396377708613323076 T^{6} -$$$$30\!\cdots\!92$$$$T^{7} +$$$$62\!\cdots\!56$$$$T^{8} -$$$$81\!\cdots\!54$$$$T^{9} +$$$$22\!\cdots\!32$$$$T^{10} -$$$$17\!\cdots\!30$$$$T^{11} +$$$$72\!\cdots\!51$$$$T^{12} -$$$$17\!\cdots\!30$$$$p^{3} T^{13} +$$$$22\!\cdots\!32$$$$p^{6} T^{14} -$$$$81\!\cdots\!54$$$$p^{9} T^{15} +$$$$62\!\cdots\!56$$$$p^{12} T^{16} -$$$$30\!\cdots\!92$$$$p^{15} T^{17} + 1396377708613323076 p^{18} T^{18} - 734700728026800 p^{21} T^{19} + 2240105622237 p^{24} T^{20} - 1020875822 p^{27} T^{21} + 2227497 p^{30} T^{22} - 624 p^{33} T^{23} + p^{36} T^{24}$$
71 $$1 + 2802 T + 5337834 T^{2} + 7404701484 T^{3} + 8651498332659 T^{4} + 8740808299670118 T^{5} + 7989057437172554298 T^{6} +$$$$66\!\cdots\!52$$$$T^{7} +$$$$51\!\cdots\!71$$$$T^{8} +$$$$37\!\cdots\!02$$$$T^{9} +$$$$25\!\cdots\!20$$$$T^{10} +$$$$16\!\cdots\!84$$$$T^{11} +$$$$10\!\cdots\!89$$$$T^{12} +$$$$16\!\cdots\!84$$$$p^{3} T^{13} +$$$$25\!\cdots\!20$$$$p^{6} T^{14} +$$$$37\!\cdots\!02$$$$p^{9} T^{15} +$$$$51\!\cdots\!71$$$$p^{12} T^{16} +$$$$66\!\cdots\!52$$$$p^{15} T^{17} + 7989057437172554298 p^{18} T^{18} + 8740808299670118 p^{21} T^{19} + 8651498332659 p^{24} T^{20} + 7404701484 p^{27} T^{21} + 5337834 p^{30} T^{22} + 2802 p^{33} T^{23} + p^{36} T^{24}$$
73 $$1 + 726 T + 2680023 T^{2} + 1620396282 T^{3} + 3655194247101 T^{4} + 1949122934768184 T^{5} + 3380858402717245766 T^{6} +$$$$16\!\cdots\!78$$$$T^{7} +$$$$23\!\cdots\!90$$$$T^{8} +$$$$10\!\cdots\!28$$$$T^{9} +$$$$12\!\cdots\!42$$$$T^{10} +$$$$49\!\cdots\!64$$$$T^{11} +$$$$75\!\cdots\!51$$$$p T^{12} +$$$$49\!\cdots\!64$$$$p^{3} T^{13} +$$$$12\!\cdots\!42$$$$p^{6} T^{14} +$$$$10\!\cdots\!28$$$$p^{9} T^{15} +$$$$23\!\cdots\!90$$$$p^{12} T^{16} +$$$$16\!\cdots\!78$$$$p^{15} T^{17} + 3380858402717245766 p^{18} T^{18} + 1949122934768184 p^{21} T^{19} + 3655194247101 p^{24} T^{20} + 1620396282 p^{27} T^{21} + 2680023 p^{30} T^{22} + 726 p^{33} T^{23} + p^{36} T^{24}$$
79 $$1 - 444 T + 5072793 T^{2} - 1964463704 T^{3} + 12036963512007 T^{4} - 4076410208752380 T^{5} + 17743948207785774412 T^{6} -$$$$52\!\cdots\!64$$$$T^{7} +$$$$18\!\cdots\!41$$$$T^{8} -$$$$47\!\cdots\!40$$$$T^{9} +$$$$13\!\cdots\!71$$$$T^{10} -$$$$31\!\cdots\!72$$$$T^{11} +$$$$77\!\cdots\!46$$$$T^{12} -$$$$31\!\cdots\!72$$$$p^{3} T^{13} +$$$$13\!\cdots\!71$$$$p^{6} T^{14} -$$$$47\!\cdots\!40$$$$p^{9} T^{15} +$$$$18\!\cdots\!41$$$$p^{12} T^{16} -$$$$52\!\cdots\!64$$$$p^{15} T^{17} + 17743948207785774412 p^{18} T^{18} - 4076410208752380 p^{21} T^{19} + 12036963512007 p^{24} T^{20} - 1964463704 p^{27} T^{21} + 5072793 p^{30} T^{22} - 444 p^{33} T^{23} + p^{36} T^{24}$$
83 $$1 - 672 T + 4281114 T^{2} - 2904905872 T^{3} + 9248883258969 T^{4} - 6065735288241552 T^{5} + 13237198106428122734 T^{6} -$$$$81\!\cdots\!04$$$$T^{7} +$$$$13\!\cdots\!82$$$$T^{8} -$$$$79\!\cdots\!68$$$$T^{9} +$$$$11\!\cdots\!30$$$$T^{10} -$$$$58\!\cdots\!12$$$$T^{11} +$$$$72\!\cdots\!57$$$$T^{12} -$$$$58\!\cdots\!12$$$$p^{3} T^{13} +$$$$11\!\cdots\!30$$$$p^{6} T^{14} -$$$$79\!\cdots\!68$$$$p^{9} T^{15} +$$$$13\!\cdots\!82$$$$p^{12} T^{16} -$$$$81\!\cdots\!04$$$$p^{15} T^{17} + 13237198106428122734 p^{18} T^{18} - 6065735288241552 p^{21} T^{19} + 9248883258969 p^{24} T^{20} - 2904905872 p^{27} T^{21} + 4281114 p^{30} T^{22} - 672 p^{33} T^{23} + p^{36} T^{24}$$
89 $$1 + 906 T + 3296823 T^{2} + 1683077954 T^{3} + 4256546856249 T^{4} + 876481176210408 T^{5} + 3114301150610841606 T^{6} -$$$$24\!\cdots\!86$$$$T^{7} +$$$$15\!\cdots\!58$$$$T^{8} -$$$$59\!\cdots\!64$$$$T^{9} +$$$$69\!\cdots\!66$$$$T^{10} -$$$$47\!\cdots\!40$$$$T^{11} +$$$$37\!\cdots\!23$$$$T^{12} -$$$$47\!\cdots\!40$$$$p^{3} T^{13} +$$$$69\!\cdots\!66$$$$p^{6} T^{14} -$$$$59\!\cdots\!64$$$$p^{9} T^{15} +$$$$15\!\cdots\!58$$$$p^{12} T^{16} -$$$$24\!\cdots\!86$$$$p^{15} T^{17} + 3114301150610841606 p^{18} T^{18} + 876481176210408 p^{21} T^{19} + 4256546856249 p^{24} T^{20} + 1683077954 p^{27} T^{21} + 3296823 p^{30} T^{22} + 906 p^{33} T^{23} + p^{36} T^{24}$$
97 $$1 - 3246 T + 11906295 T^{2} - 26951325746 T^{3} + 59212806180915 T^{4} - 104655424107839712 T^{5} +$$$$17\!\cdots\!52$$$$T^{6} -$$$$25\!\cdots\!04$$$$T^{7} +$$$$34\!\cdots\!41$$$$T^{8} -$$$$42\!\cdots\!90$$$$T^{9} +$$$$48\!\cdots\!29$$$$T^{10} -$$$$51\!\cdots\!22$$$$T^{11} +$$$$51\!\cdots\!30$$$$T^{12} -$$$$51\!\cdots\!22$$$$p^{3} T^{13} +$$$$48\!\cdots\!29$$$$p^{6} T^{14} -$$$$42\!\cdots\!90$$$$p^{9} T^{15} +$$$$34\!\cdots\!41$$$$p^{12} T^{16} -$$$$25\!\cdots\!04$$$$p^{15} T^{17} +$$$$17\!\cdots\!52$$$$p^{18} T^{18} - 104655424107839712 p^{21} T^{19} + 59212806180915 p^{24} T^{20} - 26951325746 p^{27} T^{21} + 11906295 p^{30} T^{22} - 3246 p^{33} T^{23} + p^{36} T^{24}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$