# Properties

 Degree $28$ Conductor $7.206\times 10^{16}$ Sign $1$ Motivic weight $4$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·2-s − 2·3-s − 2·4-s − 2·5-s + 4·6-s − 4·7-s − 24·8-s + 2·9-s + 4·10-s + 94·11-s + 4·12-s − 2·13-s + 8·14-s + 4·15-s + 20·16-s − 4·17-s − 4·18-s − 706·19-s + 4·20-s + 8·21-s − 188·22-s + 1.14e3·23-s + 48·24-s + 2·25-s + 4·26-s − 610·27-s + 8·28-s + ⋯
 L(s)  = 1 − 1/2·2-s − 2/9·3-s − 1/8·4-s − 0.0799·5-s + 1/9·6-s − 0.0816·7-s − 3/8·8-s + 2/81·9-s + 1/25·10-s + 0.776·11-s + 1/36·12-s − 0.0118·13-s + 2/49·14-s + 0.0177·15-s + 5/64·16-s − 0.0138·17-s − 0.0123·18-s − 1.95·19-s + 0.00999·20-s + 0.0181·21-s − 0.388·22-s + 2.17·23-s + 1/12·24-s + 0.00319·25-s + 0.00591·26-s − 0.836·27-s + 1/98·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56}\right)^{s/2} \, \Gamma_{\C}(s+2)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$28$$ Conductor: $$2^{56}$$ Sign: $1$ Motivic weight: $$4$$ Character: induced by $\chi_{16} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(28,\ 2^{56} ,\ ( \ : [2]^{14} ),\ 1 )$$

## Particular Values

 $$L(\frac{5}{2})$$ $$\approx$$ $$1.28480$$ $$L(\frac12)$$ $$\approx$$ $$1.28480$$ $$L(3)$$ 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 T + 3 p T^{2} + 5 p^{3} T^{3} + 15 p^{3} T^{4} - 11 p^{5} T^{5} - 45 p^{6} T^{6} + 5 p^{9} T^{7} - 45 p^{10} T^{8} - 11 p^{13} T^{9} + 15 p^{15} T^{10} + 5 p^{19} T^{11} + 3 p^{21} T^{12} + p^{25} T^{13} + p^{28} T^{14}$$
good3 $$1 + 2 T + 2 T^{2} + 610 T^{3} - 4613 T^{4} + 31732 T^{5} + 258740 T^{6} + 704956 p T^{7} + 7816417 p^{2} T^{8} + 4734970 p^{3} T^{9} + 12765802 p^{5} T^{10} + 91364362 p^{5} T^{11} + 11505403 p^{8} T^{12} + 372243416 p^{8} T^{13} + 3151621976 p^{8} T^{14} + 372243416 p^{12} T^{15} + 11505403 p^{16} T^{16} + 91364362 p^{17} T^{17} + 12765802 p^{21} T^{18} + 4734970 p^{23} T^{19} + 7816417 p^{26} T^{20} + 704956 p^{29} T^{21} + 258740 p^{32} T^{22} + 31732 p^{36} T^{23} - 4613 p^{40} T^{24} + 610 p^{44} T^{25} + 2 p^{48} T^{26} + 2 p^{52} T^{27} + p^{56} T^{28}$$
5 $$1 + 2 T + 2 T^{2} + 3938 T^{3} - 18913 p T^{4} - 9371916 T^{5} - 2160156 p T^{6} - 6597908556 T^{7} - 151886931511 T^{8} + 563896366366 T^{9} + 18344031697054 T^{10} - 1149733079121218 T^{11} + 112600777623819 p^{4} T^{12} + 53569344141254552 p^{2} T^{13} + 14585682673141755608 T^{14} + 53569344141254552 p^{6} T^{15} + 112600777623819 p^{12} T^{16} - 1149733079121218 p^{12} T^{17} + 18344031697054 p^{16} T^{18} + 563896366366 p^{20} T^{19} - 151886931511 p^{24} T^{20} - 6597908556 p^{28} T^{21} - 2160156 p^{33} T^{22} - 9371916 p^{36} T^{23} - 18913 p^{41} T^{24} + 3938 p^{44} T^{25} + 2 p^{48} T^{26} + 2 p^{52} T^{27} + p^{56} T^{28}$$
7 $$( 1 + 2 T + 8235 T^{2} + 64404 T^{3} + 38860249 T^{4} + 447351454 T^{5} + 125587217723 T^{6} + 196872839848 p T^{7} + 125587217723 p^{4} T^{8} + 447351454 p^{8} T^{9} + 38860249 p^{12} T^{10} + 64404 p^{16} T^{11} + 8235 p^{20} T^{12} + 2 p^{24} T^{13} + p^{28} T^{14} )^{2}$$
11 $$1 - 94 T + 4418 T^{2} + 965570 T^{3} - 470088133 T^{4} + 283723804 p T^{5} + 2249641670708 T^{6} - 796542815073292 T^{7} + 97369777194709097 T^{8} + 5317394377195027646 T^{9} -$$$$63\!\cdots\!78$$$$T^{10} +$$$$14\!\cdots\!86$$$$T^{11} -$$$$19\!\cdots\!01$$$$p^{2} T^{12} -$$$$20\!\cdots\!32$$$$p T^{13} +$$$$18\!\cdots\!92$$$$T^{14} -$$$$20\!\cdots\!32$$$$p^{5} T^{15} -$$$$19\!\cdots\!01$$$$p^{10} T^{16} +$$$$14\!\cdots\!86$$$$p^{12} T^{17} -$$$$63\!\cdots\!78$$$$p^{16} T^{18} + 5317394377195027646 p^{20} T^{19} + 97369777194709097 p^{24} T^{20} - 796542815073292 p^{28} T^{21} + 2249641670708 p^{32} T^{22} + 283723804 p^{37} T^{23} - 470088133 p^{40} T^{24} + 965570 p^{44} T^{25} + 4418 p^{48} T^{26} - 94 p^{52} T^{27} + p^{56} T^{28}$$
13 $$1 + 2 T + 2 T^{2} + 6883234 T^{3} + 1464853339 T^{4} + 65707775476 T^{5} + 1832149307204 p T^{6} + 16199073445624116 T^{7} + 1142495439970904649 T^{8} +$$$$10\!\cdots\!70$$$$T^{9} +$$$$78\!\cdots\!46$$$$T^{10} +$$$$14\!\cdots\!90$$$$T^{11} +$$$$74\!\cdots\!35$$$$T^{12} +$$$$22\!\cdots\!32$$$$T^{13} +$$$$94\!\cdots\!60$$$$T^{14} +$$$$22\!\cdots\!32$$$$p^{4} T^{15} +$$$$74\!\cdots\!35$$$$p^{8} T^{16} +$$$$14\!\cdots\!90$$$$p^{12} T^{17} +$$$$78\!\cdots\!46$$$$p^{16} T^{18} +$$$$10\!\cdots\!70$$$$p^{20} T^{19} + 1142495439970904649 p^{24} T^{20} + 16199073445624116 p^{28} T^{21} + 1832149307204 p^{33} T^{22} + 65707775476 p^{36} T^{23} + 1464853339 p^{40} T^{24} + 6883234 p^{44} T^{25} + 2 p^{48} T^{26} + 2 p^{52} T^{27} + p^{56} T^{28}$$
17 $$( 1 + 2 T + 333755 T^{2} - 12776716 T^{3} + 3318766585 p T^{4} - 174198961346 p T^{5} + 6454907058757691 T^{6} - 326099715157486120 T^{7} + 6454907058757691 p^{4} T^{8} - 174198961346 p^{9} T^{9} + 3318766585 p^{13} T^{10} - 12776716 p^{16} T^{11} + 333755 p^{20} T^{12} + 2 p^{24} T^{13} + p^{28} T^{14} )^{2}$$
19 $$1 + 706 T + 249218 T^{2} + 101381538 T^{3} + 32599242619 T^{4} + 3617133340788 T^{5} - 431513784802892 T^{6} - 960761925731564364 T^{7} -$$$$71\!\cdots\!35$$$$T^{8} -$$$$22\!\cdots\!14$$$$T^{9} -$$$$57\!\cdots\!66$$$$T^{10} -$$$$18\!\cdots\!22$$$$T^{11} -$$$$19\!\cdots\!61$$$$T^{12} +$$$$38\!\cdots\!56$$$$T^{13} +$$$$11\!\cdots\!48$$$$T^{14} +$$$$38\!\cdots\!56$$$$p^{4} T^{15} -$$$$19\!\cdots\!61$$$$p^{8} T^{16} -$$$$18\!\cdots\!22$$$$p^{12} T^{17} -$$$$57\!\cdots\!66$$$$p^{16} T^{18} -$$$$22\!\cdots\!14$$$$p^{20} T^{19} -$$$$71\!\cdots\!35$$$$p^{24} T^{20} - 960761925731564364 p^{28} T^{21} - 431513784802892 p^{32} T^{22} + 3617133340788 p^{36} T^{23} + 32599242619 p^{40} T^{24} + 101381538 p^{44} T^{25} + 249218 p^{48} T^{26} + 706 p^{52} T^{27} + p^{56} T^{28}$$
23 $$( 1 - 574 T + 1024043 T^{2} - 635922028 T^{3} + 551773439769 T^{4} - 314763003369506 T^{5} + 213700110666561659 T^{6} -$$$$10\!\cdots\!08$$$$T^{7} + 213700110666561659 p^{4} T^{8} - 314763003369506 p^{8} T^{9} + 551773439769 p^{12} T^{10} - 635922028 p^{16} T^{11} + 1024043 p^{20} T^{12} - 574 p^{24} T^{13} + p^{28} T^{14} )^{2}$$
29 $$1 - 862 T + 371522 T^{2} - 1045006654 T^{3} + 1721779716827 T^{4} - 691721668187596 T^{5} + 502604487474844916 T^{6} -$$$$98\!\cdots\!28$$$$T^{7} +$$$$75\!\cdots\!49$$$$T^{8} -$$$$31\!\cdots\!74$$$$T^{9} +$$$$39\!\cdots\!98$$$$T^{10} -$$$$15\!\cdots\!86$$$$p T^{11} +$$$$34\!\cdots\!87$$$$T^{12} -$$$$29\!\cdots\!64$$$$T^{13} +$$$$24\!\cdots\!96$$$$T^{14} -$$$$29\!\cdots\!64$$$$p^{4} T^{15} +$$$$34\!\cdots\!87$$$$p^{8} T^{16} -$$$$15\!\cdots\!86$$$$p^{13} T^{17} +$$$$39\!\cdots\!98$$$$p^{16} T^{18} -$$$$31\!\cdots\!74$$$$p^{20} T^{19} +$$$$75\!\cdots\!49$$$$p^{24} T^{20} -$$$$98\!\cdots\!28$$$$p^{28} T^{21} + 502604487474844916 p^{32} T^{22} - 691721668187596 p^{36} T^{23} + 1721779716827 p^{40} T^{24} - 1045006654 p^{44} T^{25} + 371522 p^{48} T^{26} - 862 p^{52} T^{27} + p^{56} T^{28}$$
31 $$1 - 6904334 T^{2} + 24182883262811 T^{4} - 57459081771770667372 T^{6} +$$$$10\!\cdots\!69$$$$T^{8} -$$$$14\!\cdots\!50$$$$T^{10} +$$$$57\!\cdots\!93$$$$p T^{12} -$$$$17\!\cdots\!20$$$$T^{14} +$$$$57\!\cdots\!93$$$$p^{9} T^{16} -$$$$14\!\cdots\!50$$$$p^{16} T^{18} +$$$$10\!\cdots\!69$$$$p^{24} T^{20} - 57459081771770667372 p^{32} T^{22} + 24182883262811 p^{40} T^{24} - 6904334 p^{48} T^{26} + p^{56} T^{28}$$
37 $$1 + 1826 T + 1667138 T^{2} + 4976934274 T^{3} + 7030206539163 T^{4} + 1716272691299380 T^{5} + 102662887807662052 p T^{6} +$$$$11\!\cdots\!60$$$$T^{7} -$$$$10\!\cdots\!63$$$$T^{8} -$$$$16\!\cdots\!22$$$$T^{9} +$$$$49\!\cdots\!50$$$$p T^{10} -$$$$21\!\cdots\!18$$$$T^{11} -$$$$10\!\cdots\!97$$$$T^{12} -$$$$14\!\cdots\!60$$$$T^{13} -$$$$67\!\cdots\!04$$$$T^{14} -$$$$14\!\cdots\!60$$$$p^{4} T^{15} -$$$$10\!\cdots\!97$$$$p^{8} T^{16} -$$$$21\!\cdots\!18$$$$p^{12} T^{17} +$$$$49\!\cdots\!50$$$$p^{17} T^{18} -$$$$16\!\cdots\!22$$$$p^{20} T^{19} -$$$$10\!\cdots\!63$$$$p^{24} T^{20} +$$$$11\!\cdots\!60$$$$p^{28} T^{21} + 102662887807662052 p^{33} T^{22} + 1716272691299380 p^{36} T^{23} + 7030206539163 p^{40} T^{24} + 4976934274 p^{44} T^{25} + 1667138 p^{48} T^{26} + 1826 p^{52} T^{27} + p^{56} T^{28}$$
41 $$1 - 24523982 T^{2} + 302442312166171 T^{4} -$$$$24\!\cdots\!76$$$$T^{6} +$$$$14\!\cdots\!01$$$$T^{8} -$$$$70\!\cdots\!70$$$$T^{10} +$$$$27\!\cdots\!51$$$$T^{12} -$$$$84\!\cdots\!88$$$$T^{14} +$$$$27\!\cdots\!51$$$$p^{8} T^{16} -$$$$70\!\cdots\!70$$$$p^{16} T^{18} +$$$$14\!\cdots\!01$$$$p^{24} T^{20} -$$$$24\!\cdots\!76$$$$p^{32} T^{22} + 302442312166171 p^{40} T^{24} - 24523982 p^{48} T^{26} + p^{56} T^{28}$$
43 $$1 - 1694 T + 1434818 T^{2} - 14278395262 T^{3} + 44454402050619 T^{4} - 13474016894363980 T^{5} + 60977294006539554100 T^{6} -$$$$38\!\cdots\!44$$$$T^{7} +$$$$23\!\cdots\!81$$$$T^{8} +$$$$78\!\cdots\!82$$$$T^{9} +$$$$12\!\cdots\!62$$$$T^{10} -$$$$18\!\cdots\!22$$$$T^{11} -$$$$91\!\cdots\!45$$$$T^{12} +$$$$10\!\cdots\!08$$$$T^{13} +$$$$16\!\cdots\!52$$$$T^{14} +$$$$10\!\cdots\!08$$$$p^{4} T^{15} -$$$$91\!\cdots\!45$$$$p^{8} T^{16} -$$$$18\!\cdots\!22$$$$p^{12} T^{17} +$$$$12\!\cdots\!62$$$$p^{16} T^{18} +$$$$78\!\cdots\!82$$$$p^{20} T^{19} +$$$$23\!\cdots\!81$$$$p^{24} T^{20} -$$$$38\!\cdots\!44$$$$p^{28} T^{21} + 60977294006539554100 p^{32} T^{22} - 13474016894363980 p^{36} T^{23} + 44454402050619 p^{40} T^{24} - 14278395262 p^{44} T^{25} + 1434818 p^{48} T^{26} - 1694 p^{52} T^{27} + p^{56} T^{28}$$
47 $$1 - 51887758 T^{2} + 1309844227745755 T^{4} -$$$$45\!\cdots\!48$$$$p T^{6} +$$$$24\!\cdots\!09$$$$T^{8} -$$$$21\!\cdots\!30$$$$T^{10} +$$$$15\!\cdots\!39$$$$T^{12} -$$$$82\!\cdots\!04$$$$T^{14} +$$$$15\!\cdots\!39$$$$p^{8} T^{16} -$$$$21\!\cdots\!30$$$$p^{16} T^{18} +$$$$24\!\cdots\!09$$$$p^{24} T^{20} -$$$$45\!\cdots\!48$$$$p^{33} T^{22} + 1309844227745755 p^{40} T^{24} - 51887758 p^{48} T^{26} + p^{56} T^{28}$$
53 $$1 + 482 T + 116162 T^{2} - 5558326078 T^{3} + 43583027341595 T^{4} + 155411473123116980 T^{5} + 85293132817975457012 T^{6} +$$$$11\!\cdots\!76$$$$T^{7} +$$$$35\!\cdots\!13$$$$T^{8} -$$$$92\!\cdots\!94$$$$T^{9} +$$$$12\!\cdots\!18$$$$T^{10} +$$$$13\!\cdots\!10$$$$T^{11} +$$$$28\!\cdots\!31$$$$T^{12} +$$$$18\!\cdots\!24$$$$T^{13} +$$$$12\!\cdots\!68$$$$T^{14} +$$$$18\!\cdots\!24$$$$p^{4} T^{15} +$$$$28\!\cdots\!31$$$$p^{8} T^{16} +$$$$13\!\cdots\!10$$$$p^{12} T^{17} +$$$$12\!\cdots\!18$$$$p^{16} T^{18} -$$$$92\!\cdots\!94$$$$p^{20} T^{19} +$$$$35\!\cdots\!13$$$$p^{24} T^{20} +$$$$11\!\cdots\!76$$$$p^{28} T^{21} + 85293132817975457012 p^{32} T^{22} + 155411473123116980 p^{36} T^{23} + 43583027341595 p^{40} T^{24} - 5558326078 p^{44} T^{25} + 116162 p^{48} T^{26} + 482 p^{52} T^{27} + p^{56} T^{28}$$
59 $$1 + 2786 T + 3880898 T^{2} + 95235375746 T^{3} + 282835430943931 T^{4} - 951817240129082700 T^{5} +$$$$78\!\cdots\!20$$$$T^{6} -$$$$10\!\cdots\!24$$$$T^{7} -$$$$10\!\cdots\!59$$$$T^{8} -$$$$16\!\cdots\!78$$$$T^{9} +$$$$95\!\cdots\!50$$$$T^{10} -$$$$27\!\cdots\!94$$$$T^{11} +$$$$66\!\cdots\!23$$$$T^{12} +$$$$40\!\cdots\!76$$$$T^{13} +$$$$34\!\cdots\!76$$$$T^{14} +$$$$40\!\cdots\!76$$$$p^{4} T^{15} +$$$$66\!\cdots\!23$$$$p^{8} T^{16} -$$$$27\!\cdots\!94$$$$p^{12} T^{17} +$$$$95\!\cdots\!50$$$$p^{16} T^{18} -$$$$16\!\cdots\!78$$$$p^{20} T^{19} -$$$$10\!\cdots\!59$$$$p^{24} T^{20} -$$$$10\!\cdots\!24$$$$p^{28} T^{21} +$$$$78\!\cdots\!20$$$$p^{32} T^{22} - 951817240129082700 p^{36} T^{23} + 282835430943931 p^{40} T^{24} + 95235375746 p^{44} T^{25} + 3880898 p^{48} T^{26} + 2786 p^{52} T^{27} + p^{56} T^{28}$$
61 $$1 + 3778 T + 7136642 T^{2} + 13584988130 T^{3} + 15885658886619 T^{4} - 322904339201283724 T^{5} -$$$$12\!\cdots\!20$$$$T^{6} -$$$$11\!\cdots\!80$$$$T^{7} -$$$$34\!\cdots\!59$$$$T^{8} -$$$$57\!\cdots\!82$$$$T^{9} -$$$$99\!\cdots\!50$$$$p T^{10} -$$$$13\!\cdots\!70$$$$T^{11} +$$$$15\!\cdots\!55$$$$T^{12} +$$$$19\!\cdots\!60$$$$T^{13} +$$$$92\!\cdots\!08$$$$T^{14} +$$$$19\!\cdots\!60$$$$p^{4} T^{15} +$$$$15\!\cdots\!55$$$$p^{8} T^{16} -$$$$13\!\cdots\!70$$$$p^{12} T^{17} -$$$$99\!\cdots\!50$$$$p^{17} T^{18} -$$$$57\!\cdots\!82$$$$p^{20} T^{19} -$$$$34\!\cdots\!59$$$$p^{24} T^{20} -$$$$11\!\cdots\!80$$$$p^{28} T^{21} -$$$$12\!\cdots\!20$$$$p^{32} T^{22} - 322904339201283724 p^{36} T^{23} + 15885658886619 p^{40} T^{24} + 13584988130 p^{44} T^{25} + 7136642 p^{48} T^{26} + 3778 p^{52} T^{27} + p^{56} T^{28}$$
67 $$1 - 7998 T + 31984002 T^{2} - 47670849246 T^{3} - 439076236005637 T^{4} + 648765759780793716 T^{5} +$$$$99\!\cdots\!64$$$$T^{6} -$$$$93\!\cdots\!52$$$$T^{7} +$$$$28\!\cdots\!21$$$$T^{8} +$$$$69\!\cdots\!70$$$$T^{9} -$$$$55\!\cdots\!14$$$$T^{10} +$$$$21\!\cdots\!74$$$$T^{11} -$$$$39\!\cdots\!49$$$$T^{12} +$$$$17\!\cdots\!60$$$$T^{13} -$$$$39\!\cdots\!76$$$$T^{14} +$$$$17\!\cdots\!60$$$$p^{4} T^{15} -$$$$39\!\cdots\!49$$$$p^{8} T^{16} +$$$$21\!\cdots\!74$$$$p^{12} T^{17} -$$$$55\!\cdots\!14$$$$p^{16} T^{18} +$$$$69\!\cdots\!70$$$$p^{20} T^{19} +$$$$28\!\cdots\!21$$$$p^{24} T^{20} -$$$$93\!\cdots\!52$$$$p^{28} T^{21} +$$$$99\!\cdots\!64$$$$p^{32} T^{22} + 648765759780793716 p^{36} T^{23} - 439076236005637 p^{40} T^{24} - 47670849246 p^{44} T^{25} + 31984002 p^{48} T^{26} - 7998 p^{52} T^{27} + p^{56} T^{28}$$
71 $$( 1 - 9982 T + 145017323 T^{2} - 1165310044396 T^{3} + 10014081489420185 T^{4} - 63641985337531169890 T^{5} +$$$$40\!\cdots\!59$$$$T^{6} -$$$$20\!\cdots\!72$$$$T^{7} +$$$$40\!\cdots\!59$$$$p^{4} T^{8} - 63641985337531169890 p^{8} T^{9} + 10014081489420185 p^{12} T^{10} - 1165310044396 p^{16} T^{11} + 145017323 p^{20} T^{12} - 9982 p^{24} T^{13} + p^{28} T^{14} )^{2}$$
73 $$1 - 168573838 T^{2} + 13353714116727067 T^{4} -$$$$70\!\cdots\!96$$$$T^{6} +$$$$30\!\cdots\!57$$$$T^{8} -$$$$11\!\cdots\!98$$$$T^{10} +$$$$39\!\cdots\!07$$$$T^{12} -$$$$11\!\cdots\!44$$$$T^{14} +$$$$39\!\cdots\!07$$$$p^{8} T^{16} -$$$$11\!\cdots\!98$$$$p^{16} T^{18} +$$$$30\!\cdots\!57$$$$p^{24} T^{20} -$$$$70\!\cdots\!96$$$$p^{32} T^{22} + 13353714116727067 p^{40} T^{24} - 168573838 p^{48} T^{26} + p^{56} T^{28}$$
79 $$1 - 364033678 T^{2} + 65454647116587227 T^{4} -$$$$76\!\cdots\!56$$$$T^{6} +$$$$65\!\cdots\!81$$$$T^{8} -$$$$43\!\cdots\!10$$$$T^{10} +$$$$23\!\cdots\!67$$$$T^{12} -$$$$99\!\cdots\!88$$$$T^{14} +$$$$23\!\cdots\!67$$$$p^{8} T^{16} -$$$$43\!\cdots\!10$$$$p^{16} T^{18} +$$$$65\!\cdots\!81$$$$p^{24} T^{20} -$$$$76\!\cdots\!56$$$$p^{32} T^{22} + 65454647116587227 p^{40} T^{24} - 364033678 p^{48} T^{26} + p^{56} T^{28}$$
83 $$1 + 17282 T + 149333762 T^{2} + 1088719641698 T^{3} + 16964332297412731 T^{4} +$$$$21\!\cdots\!96$$$$T^{5} +$$$$17\!\cdots\!52$$$$T^{6} +$$$$12\!\cdots\!12$$$$T^{7} +$$$$11\!\cdots\!61$$$$T^{8} +$$$$11\!\cdots\!62$$$$T^{9} +$$$$90\!\cdots\!74$$$$T^{10} +$$$$61\!\cdots\!82$$$$T^{11} +$$$$44\!\cdots\!67$$$$T^{12} +$$$$35\!\cdots\!48$$$$T^{13} +$$$$26\!\cdots\!76$$$$T^{14} +$$$$35\!\cdots\!48$$$$p^{4} T^{15} +$$$$44\!\cdots\!67$$$$p^{8} T^{16} +$$$$61\!\cdots\!82$$$$p^{12} T^{17} +$$$$90\!\cdots\!74$$$$p^{16} T^{18} +$$$$11\!\cdots\!62$$$$p^{20} T^{19} +$$$$11\!\cdots\!61$$$$p^{24} T^{20} +$$$$12\!\cdots\!12$$$$p^{28} T^{21} +$$$$17\!\cdots\!52$$$$p^{32} T^{22} +$$$$21\!\cdots\!96$$$$p^{36} T^{23} + 16964332297412731 p^{40} T^{24} + 1088719641698 p^{44} T^{25} + 149333762 p^{48} T^{26} + 17282 p^{52} T^{27} + p^{56} T^{28}$$
89 $$1 - 548528910 T^{2} + 149200943223060123 T^{4} -$$$$26\!\cdots\!16$$$$T^{6} +$$$$35\!\cdots\!49$$$$T^{8} -$$$$37\!\cdots\!50$$$$T^{10} +$$$$31\!\cdots\!11$$$$T^{12} -$$$$21\!\cdots\!00$$$$T^{14} +$$$$31\!\cdots\!11$$$$p^{8} T^{16} -$$$$37\!\cdots\!50$$$$p^{16} T^{18} +$$$$35\!\cdots\!49$$$$p^{24} T^{20} -$$$$26\!\cdots\!16$$$$p^{32} T^{22} + 149200943223060123 p^{40} T^{24} - 548528910 p^{48} T^{26} + p^{56} T^{28}$$
97 $$( 1 + 2 T + 387850619 T^{2} + 251760181236 T^{3} + 75114732161345545 T^{4} + 73269666487293981214 T^{5} +$$$$94\!\cdots\!67$$$$T^{6} +$$$$89\!\cdots\!44$$$$T^{7} +$$$$94\!\cdots\!67$$$$p^{4} T^{8} + 73269666487293981214 p^{8} T^{9} + 75114732161345545 p^{12} T^{10} + 251760181236 p^{16} T^{11} + 387850619 p^{20} T^{12} + 2 p^{24} T^{13} + p^{28} T^{14} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{28} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$