# Properties

 Label 28-74e14-1.1-c4e14-0-0 Degree $28$ Conductor $1.477\times 10^{26}$ Sign $1$ Analytic cond. $2.34836\times 10^{12}$ Root an. cond. $2.76575$ Motivic weight $4$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 28·2-s + 392·4-s − 12·5-s + 48·7-s − 3.58e3·8-s + 394·9-s + 336·10-s − 56·13-s − 1.34e3·14-s + 2.32e4·16-s − 348·17-s − 1.10e4·18-s − 184·19-s − 4.70e3·20-s − 502·23-s + 72·25-s + 1.56e3·26-s + 1.88e4·28-s − 474·29-s − 630·31-s − 1.07e5·32-s + 9.74e3·34-s − 576·35-s + 1.54e5·36-s − 2.54e3·37-s + 5.15e3·38-s + 4.30e4·40-s + ⋯
 L(s)  = 1 − 7·2-s + 49/2·4-s − 0.479·5-s + 0.979·7-s − 56·8-s + 4.86·9-s + 3.35·10-s − 0.331·13-s − 6.85·14-s + 91·16-s − 1.20·17-s − 34.0·18-s − 0.509·19-s − 11.7·20-s − 0.948·23-s + 0.115·25-s + 2.31·26-s + 24·28-s − 0.563·29-s − 0.655·31-s − 105·32-s + 8.42·34-s − 0.470·35-s + 119.·36-s − 1.85·37-s + 3.56·38-s + 26.8·40-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{14} \cdot 37^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{14} \cdot 37^{14}\right)^{s/2} \, \Gamma_{\C}(s+2)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$28$$ Conductor: $$2^{14} \cdot 37^{14}$$ Sign: $1$ Analytic conductor: $$2.34836\times 10^{12}$$ Root analytic conductor: $$2.76575$$ Motivic weight: $$4$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{74} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(28,\ 2^{14} \cdot 37^{14} ,\ ( \ : [2]^{14} ),\ 1 )$$

## Particular Values

 $$L(\frac{5}{2})$$ $$\approx$$ $$7.155446577\times10^{-8}$$ $$L(\frac12)$$ $$\approx$$ $$7.155446577\times10^{-8}$$ $$L(3)$$ 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^{2} T + p^{3} T^{2} )^{7}$$
37 $$1 + 2544 T + 6957359 T^{2} + 355685792 p T^{3} + 18562000697 p^{2} T^{4} + 733759969744 p^{3} T^{5} + 31475360416911 p^{4} T^{6} + 31271431299520 p^{6} T^{7} + 31475360416911 p^{8} T^{8} + 733759969744 p^{11} T^{9} + 18562000697 p^{14} T^{10} + 355685792 p^{17} T^{11} + 6957359 p^{20} T^{12} + 2544 p^{24} T^{13} + p^{28} T^{14}$$
good3 $$1 - 394 T^{2} + 85051 T^{4} - 4489654 p T^{6} + 190915985 p^{2} T^{8} - 20563844282 p^{2} T^{10} + 216342796100 p^{4} T^{12} - 2048126398358 p^{6} T^{14} + 216342796100 p^{12} T^{16} - 20563844282 p^{18} T^{18} + 190915985 p^{26} T^{20} - 4489654 p^{33} T^{22} + 85051 p^{40} T^{24} - 394 p^{48} T^{26} + p^{56} T^{28}$$
5 $$1 + 12 T + 72 T^{2} + 2844 p T^{3} - 213098 T^{4} + 2486568 T^{5} + 146286072 T^{6} + 8780919948 T^{7} + 317380465353 T^{8} - 2574111991608 T^{9} + 105388723836216 T^{10} - 210309660295008 T^{11} - 37629164093474656 T^{12} + 2902308242516634828 T^{13} + 42727753081405713168 T^{14} + 2902308242516634828 p^{4} T^{15} - 37629164093474656 p^{8} T^{16} - 210309660295008 p^{12} T^{17} + 105388723836216 p^{16} T^{18} - 2574111991608 p^{20} T^{19} + 317380465353 p^{24} T^{20} + 8780919948 p^{28} T^{21} + 146286072 p^{32} T^{22} + 2486568 p^{36} T^{23} - 213098 p^{40} T^{24} + 2844 p^{45} T^{25} + 72 p^{48} T^{26} + 12 p^{52} T^{27} + p^{56} T^{28}$$
7 $$( 1 - 24 T + 6726 T^{2} - 51742 T^{3} + 18052794 T^{4} + 587182460 T^{5} + 21407683291 T^{6} + 2651018020764 T^{7} + 21407683291 p^{4} T^{8} + 587182460 p^{8} T^{9} + 18052794 p^{12} T^{10} - 51742 p^{16} T^{11} + 6726 p^{20} T^{12} - 24 p^{24} T^{13} + p^{28} T^{14} )^{2}$$
11 $$1 - 85002 T^{2} + 4114081755 T^{4} - 140668263843970 T^{6} + 3745236095209650585 T^{8} -$$$$81\!\cdots\!38$$$$T^{10} +$$$$13\!\cdots\!84$$$$p T^{12} -$$$$23\!\cdots\!26$$$$T^{14} +$$$$13\!\cdots\!84$$$$p^{9} T^{16} -$$$$81\!\cdots\!38$$$$p^{16} T^{18} + 3745236095209650585 p^{24} T^{20} - 140668263843970 p^{32} T^{22} + 4114081755 p^{40} T^{24} - 85002 p^{48} T^{26} + p^{56} T^{28}$$
13 $$1 + 56 T + 1568 T^{2} - 2724856 T^{3} - 3165213706 T^{4} + 5539540132 T^{5} + 8985689448768 T^{6} + 12875785786747712 T^{7} + 304950936705328717 p T^{8} -$$$$36\!\cdots\!32$$$$T^{9} -$$$$33\!\cdots\!72$$$$T^{10} -$$$$17\!\cdots\!76$$$$p T^{11} -$$$$16\!\cdots\!92$$$$T^{12} +$$$$67\!\cdots\!92$$$$T^{13} +$$$$41\!\cdots\!24$$$$T^{14} +$$$$67\!\cdots\!92$$$$p^{4} T^{15} -$$$$16\!\cdots\!92$$$$p^{8} T^{16} -$$$$17\!\cdots\!76$$$$p^{13} T^{17} -$$$$33\!\cdots\!72$$$$p^{16} T^{18} -$$$$36\!\cdots\!32$$$$p^{20} T^{19} + 304950936705328717 p^{25} T^{20} + 12875785786747712 p^{28} T^{21} + 8985689448768 p^{32} T^{22} + 5539540132 p^{36} T^{23} - 3165213706 p^{40} T^{24} - 2724856 p^{44} T^{25} + 1568 p^{48} T^{26} + 56 p^{52} T^{27} + p^{56} T^{28}$$
17 $$1 + 348 T + 60552 T^{2} - 9208724 T^{3} - 20637046009 T^{4} - 5605095097528 T^{5} - 658558385148688 T^{6} - 19269798768974168 T^{7} +$$$$12\!\cdots\!65$$$$T^{8} +$$$$49\!\cdots\!68$$$$T^{9} +$$$$71\!\cdots\!48$$$$p T^{10} +$$$$42\!\cdots\!60$$$$T^{11} -$$$$16\!\cdots\!53$$$$T^{12} -$$$$47\!\cdots\!36$$$$T^{13} -$$$$14\!\cdots\!40$$$$T^{14} -$$$$47\!\cdots\!36$$$$p^{4} T^{15} -$$$$16\!\cdots\!53$$$$p^{8} T^{16} +$$$$42\!\cdots\!60$$$$p^{12} T^{17} +$$$$71\!\cdots\!48$$$$p^{17} T^{18} +$$$$49\!\cdots\!68$$$$p^{20} T^{19} +$$$$12\!\cdots\!65$$$$p^{24} T^{20} - 19269798768974168 p^{28} T^{21} - 658558385148688 p^{32} T^{22} - 5605095097528 p^{36} T^{23} - 20637046009 p^{40} T^{24} - 9208724 p^{44} T^{25} + 60552 p^{48} T^{26} + 348 p^{52} T^{27} + p^{56} T^{28}$$
19 $$1 + 184 T + 16928 T^{2} + 17619296 T^{3} + 15867618359 T^{4} + 2617590392536 T^{5} + 368249384413280 T^{6} + 626089397840613048 T^{7} +$$$$49\!\cdots\!09$$$$T^{8} -$$$$46\!\cdots\!28$$$$T^{9} -$$$$54\!\cdots\!68$$$$T^{10} +$$$$91\!\cdots\!32$$$$T^{11} +$$$$40\!\cdots\!75$$$$T^{12} +$$$$30\!\cdots\!36$$$$T^{13} +$$$$16\!\cdots\!32$$$$T^{14} +$$$$30\!\cdots\!36$$$$p^{4} T^{15} +$$$$40\!\cdots\!75$$$$p^{8} T^{16} +$$$$91\!\cdots\!32$$$$p^{12} T^{17} -$$$$54\!\cdots\!68$$$$p^{16} T^{18} -$$$$46\!\cdots\!28$$$$p^{20} T^{19} +$$$$49\!\cdots\!09$$$$p^{24} T^{20} + 626089397840613048 p^{28} T^{21} + 368249384413280 p^{32} T^{22} + 2617590392536 p^{36} T^{23} + 15867618359 p^{40} T^{24} + 17619296 p^{44} T^{25} + 16928 p^{48} T^{26} + 184 p^{52} T^{27} + p^{56} T^{28}$$
23 $$1 + 502 T + 126002 T^{2} - 179350290 T^{3} - 257807725322 T^{4} - 85939758964330 T^{5} + 5425793267471034 T^{6} + 37683286712771016358 T^{7} +$$$$23\!\cdots\!21$$$$T^{8} -$$$$42\!\cdots\!72$$$$T^{9} -$$$$49\!\cdots\!08$$$$T^{10} -$$$$40\!\cdots\!40$$$$T^{11} -$$$$73\!\cdots\!00$$$$T^{12} +$$$$87\!\cdots\!92$$$$T^{13} +$$$$60\!\cdots\!36$$$$T^{14} +$$$$87\!\cdots\!92$$$$p^{4} T^{15} -$$$$73\!\cdots\!00$$$$p^{8} T^{16} -$$$$40\!\cdots\!40$$$$p^{12} T^{17} -$$$$49\!\cdots\!08$$$$p^{16} T^{18} -$$$$42\!\cdots\!72$$$$p^{20} T^{19} +$$$$23\!\cdots\!21$$$$p^{24} T^{20} + 37683286712771016358 p^{28} T^{21} + 5425793267471034 p^{32} T^{22} - 85939758964330 p^{36} T^{23} - 257807725322 p^{40} T^{24} - 179350290 p^{44} T^{25} + 126002 p^{48} T^{26} + 502 p^{52} T^{27} + p^{56} T^{28}$$
29 $$1 + 474 T + 112338 T^{2} + 62620330 T^{3} + 276408573086 T^{4} + 286931828710502 T^{5} + 106915153390097330 T^{6} +$$$$16\!\cdots\!22$$$$T^{7} -$$$$33\!\cdots\!27$$$$T^{8} -$$$$17\!\cdots\!84$$$$T^{9} -$$$$22\!\cdots\!08$$$$T^{10} +$$$$34\!\cdots\!84$$$$T^{11} -$$$$14\!\cdots\!44$$$$T^{12} -$$$$12\!\cdots\!44$$$$T^{13} -$$$$48\!\cdots\!92$$$$T^{14} -$$$$12\!\cdots\!44$$$$p^{4} T^{15} -$$$$14\!\cdots\!44$$$$p^{8} T^{16} +$$$$34\!\cdots\!84$$$$p^{12} T^{17} -$$$$22\!\cdots\!08$$$$p^{16} T^{18} -$$$$17\!\cdots\!84$$$$p^{20} T^{19} -$$$$33\!\cdots\!27$$$$p^{24} T^{20} +$$$$16\!\cdots\!22$$$$p^{28} T^{21} + 106915153390097330 p^{32} T^{22} + 286931828710502 p^{36} T^{23} + 276408573086 p^{40} T^{24} + 62620330 p^{44} T^{25} + 112338 p^{48} T^{26} + 474 p^{52} T^{27} + p^{56} T^{28}$$
31 $$1 + 630 T + 198450 T^{2} + 217065406 T^{3} - 1229066036006 T^{4} - 1052289838784874 T^{5} - 395475748348107502 T^{6} -$$$$56\!\cdots\!38$$$$T^{7} -$$$$57\!\cdots\!27$$$$T^{8} -$$$$28\!\cdots\!04$$$$T^{9} -$$$$73\!\cdots\!72$$$$T^{10} -$$$$47\!\cdots\!04$$$$T^{11} +$$$$33\!\cdots\!48$$$$p^{2} T^{12} +$$$$33\!\cdots\!04$$$$p T^{13} +$$$$58\!\cdots\!28$$$$T^{14} +$$$$33\!\cdots\!04$$$$p^{5} T^{15} +$$$$33\!\cdots\!48$$$$p^{10} T^{16} -$$$$47\!\cdots\!04$$$$p^{12} T^{17} -$$$$73\!\cdots\!72$$$$p^{16} T^{18} -$$$$28\!\cdots\!04$$$$p^{20} T^{19} -$$$$57\!\cdots\!27$$$$p^{24} T^{20} -$$$$56\!\cdots\!38$$$$p^{28} T^{21} - 395475748348107502 p^{32} T^{22} - 1052289838784874 p^{36} T^{23} - 1229066036006 p^{40} T^{24} + 217065406 p^{44} T^{25} + 198450 p^{48} T^{26} + 630 p^{52} T^{27} + p^{56} T^{28}$$
41 $$1 - 19524398 T^{2} + 184948999818539 T^{4} -$$$$11\!\cdots\!14$$$$T^{6} +$$$$48\!\cdots\!01$$$$T^{8} -$$$$16\!\cdots\!94$$$$T^{10} +$$$$46\!\cdots\!72$$$$T^{12} -$$$$12\!\cdots\!78$$$$T^{14} +$$$$46\!\cdots\!72$$$$p^{8} T^{16} -$$$$16\!\cdots\!94$$$$p^{16} T^{18} +$$$$48\!\cdots\!01$$$$p^{24} T^{20} -$$$$11\!\cdots\!14$$$$p^{32} T^{22} + 184948999818539 p^{40} T^{24} - 19524398 p^{48} T^{26} + p^{56} T^{28}$$
43 $$1 - 1936 T + 1874048 T^{2} - 15836584 p T^{3} - 19739758413961 T^{4} + 37135213465156088 T^{5} - 34668656302741919968 T^{6} +$$$$71\!\cdots\!24$$$$T^{7} +$$$$83\!\cdots\!77$$$$T^{8} -$$$$28\!\cdots\!68$$$$T^{9} +$$$$49\!\cdots\!88$$$$T^{10} -$$$$16\!\cdots\!92$$$$T^{11} +$$$$11\!\cdots\!83$$$$T^{12} +$$$$39\!\cdots\!96$$$$T^{13} -$$$$66\!\cdots\!28$$$$T^{14} +$$$$39\!\cdots\!96$$$$p^{4} T^{15} +$$$$11\!\cdots\!83$$$$p^{8} T^{16} -$$$$16\!\cdots\!92$$$$p^{12} T^{17} +$$$$49\!\cdots\!88$$$$p^{16} T^{18} -$$$$28\!\cdots\!68$$$$p^{20} T^{19} +$$$$83\!\cdots\!77$$$$p^{24} T^{20} +$$$$71\!\cdots\!24$$$$p^{28} T^{21} - 34668656302741919968 p^{32} T^{22} + 37135213465156088 p^{36} T^{23} - 19739758413961 p^{40} T^{24} - 15836584 p^{45} T^{25} + 1874048 p^{48} T^{26} - 1936 p^{52} T^{27} + p^{56} T^{28}$$
47 $$( 1 - 2858 T + 16702730 T^{2} - 38868072582 T^{3} + 156182657105096 T^{4} - 318036430116401934 T^{5} +$$$$10\!\cdots\!77$$$$T^{6} -$$$$18\!\cdots\!08$$$$T^{7} +$$$$10\!\cdots\!77$$$$p^{4} T^{8} - 318036430116401934 p^{8} T^{9} + 156182657105096 p^{12} T^{10} - 38868072582 p^{16} T^{11} + 16702730 p^{20} T^{12} - 2858 p^{24} T^{13} + p^{28} T^{14} )^{2}$$
53 $$( 1 + 10114 T + 64251564 T^{2} + 290203277974 T^{3} + 1064908639250588 T^{4} + 3332341116188935666 T^{5} +$$$$94\!\cdots\!67$$$$T^{6} +$$$$26\!\cdots\!60$$$$T^{7} +$$$$94\!\cdots\!67$$$$p^{4} T^{8} + 3332341116188935666 p^{8} T^{9} + 1064908639250588 p^{12} T^{10} + 290203277974 p^{16} T^{11} + 64251564 p^{20} T^{12} + 10114 p^{24} T^{13} + p^{28} T^{14} )^{2}$$
59 $$1 + 4502 T + 10134002 T^{2} + 70800952390 T^{3} - 120345836546625 T^{4} - 1468558928300356636 T^{5} -$$$$28\!\cdots\!72$$$$T^{6} -$$$$18\!\cdots\!60$$$$T^{7} +$$$$17\!\cdots\!65$$$$T^{8} +$$$$31\!\cdots\!46$$$$T^{9} +$$$$65\!\cdots\!10$$$$T^{10} +$$$$41\!\cdots\!70$$$$T^{11} -$$$$31\!\cdots\!05$$$$T^{12} -$$$$64\!\cdots\!72$$$$T^{13} -$$$$12\!\cdots\!60$$$$T^{14} -$$$$64\!\cdots\!72$$$$p^{4} T^{15} -$$$$31\!\cdots\!05$$$$p^{8} T^{16} +$$$$41\!\cdots\!70$$$$p^{12} T^{17} +$$$$65\!\cdots\!10$$$$p^{16} T^{18} +$$$$31\!\cdots\!46$$$$p^{20} T^{19} +$$$$17\!\cdots\!65$$$$p^{24} T^{20} -$$$$18\!\cdots\!60$$$$p^{28} T^{21} -$$$$28\!\cdots\!72$$$$p^{32} T^{22} - 1468558928300356636 p^{36} T^{23} - 120345836546625 p^{40} T^{24} + 70800952390 p^{44} T^{25} + 10134002 p^{48} T^{26} + 4502 p^{52} T^{27} + p^{56} T^{28}$$
61 $$1 + 11906 T + 70876418 T^{2} + 203665889578 T^{3} - 394142745410454 T^{4} - 5986165856758813458 T^{5} -$$$$22\!\cdots\!34$$$$T^{6} -$$$$63\!\cdots\!30$$$$T^{7} +$$$$33\!\cdots\!33$$$$T^{8} +$$$$14\!\cdots\!60$$$$T^{9} +$$$$13\!\cdots\!72$$$$T^{10} -$$$$11\!\cdots\!52$$$$T^{11} -$$$$49\!\cdots\!44$$$$T^{12} -$$$$26\!\cdots\!40$$$$T^{13} +$$$$27\!\cdots\!76$$$$T^{14} -$$$$26\!\cdots\!40$$$$p^{4} T^{15} -$$$$49\!\cdots\!44$$$$p^{8} T^{16} -$$$$11\!\cdots\!52$$$$p^{12} T^{17} +$$$$13\!\cdots\!72$$$$p^{16} T^{18} +$$$$14\!\cdots\!60$$$$p^{20} T^{19} +$$$$33\!\cdots\!33$$$$p^{24} T^{20} -$$$$63\!\cdots\!30$$$$p^{28} T^{21} -$$$$22\!\cdots\!34$$$$p^{32} T^{22} - 5986165856758813458 p^{36} T^{23} - 394142745410454 p^{40} T^{24} + 203665889578 p^{44} T^{25} + 70876418 p^{48} T^{26} + 11906 p^{52} T^{27} + p^{56} T^{28}$$
67 $$1 - 184400600 T^{2} + 16859785332405158 T^{4} -$$$$10\!\cdots\!92$$$$T^{6} +$$$$43\!\cdots\!61$$$$T^{8} -$$$$14\!\cdots\!98$$$$T^{10} +$$$$40\!\cdots\!32$$$$T^{12} -$$$$89\!\cdots\!60$$$$T^{14} +$$$$40\!\cdots\!32$$$$p^{8} T^{16} -$$$$14\!\cdots\!98$$$$p^{16} T^{18} +$$$$43\!\cdots\!61$$$$p^{24} T^{20} -$$$$10\!\cdots\!92$$$$p^{32} T^{22} + 16859785332405158 p^{40} T^{24} - 184400600 p^{48} T^{26} + p^{56} T^{28}$$
71 $$( 1 + 5612 T + 60900748 T^{2} + 483650204234 T^{3} + 2976850829770066 T^{4} + 19760900351284778516 T^{5} +$$$$11\!\cdots\!69$$$$T^{6} +$$$$53\!\cdots\!56$$$$T^{7} +$$$$11\!\cdots\!69$$$$p^{4} T^{8} + 19760900351284778516 p^{8} T^{9} + 2976850829770066 p^{12} T^{10} + 483650204234 p^{16} T^{11} + 60900748 p^{20} T^{12} + 5612 p^{24} T^{13} + p^{28} T^{14} )^{2}$$
73 $$1 - 95040550 T^{2} + 4729797140050243 T^{4} -$$$$18\!\cdots\!34$$$$T^{6} +$$$$70\!\cdots\!77$$$$T^{8} -$$$$24\!\cdots\!26$$$$T^{10} +$$$$77\!\cdots\!16$$$$T^{12} -$$$$23\!\cdots\!18$$$$T^{14} +$$$$77\!\cdots\!16$$$$p^{8} T^{16} -$$$$24\!\cdots\!26$$$$p^{16} T^{18} +$$$$70\!\cdots\!77$$$$p^{24} T^{20} -$$$$18\!\cdots\!34$$$$p^{32} T^{22} + 4729797140050243 p^{40} T^{24} - 95040550 p^{48} T^{26} + p^{56} T^{28}$$
79 $$1 - 20488 T + 209879072 T^{2} - 1422808826512 T^{3} + 7013576890902382 T^{4} - 35007990661500345608 T^{5} +$$$$25\!\cdots\!72$$$$T^{6} -$$$$19\!\cdots\!04$$$$T^{7} +$$$$11\!\cdots\!69$$$$T^{8} -$$$$70\!\cdots\!44$$$$p T^{9} +$$$$29\!\cdots\!96$$$$T^{10} -$$$$16\!\cdots\!64$$$$T^{11} +$$$$47\!\cdots\!04$$$$T^{12} +$$$$28\!\cdots\!12$$$$T^{13} -$$$$35\!\cdots\!00$$$$T^{14} +$$$$28\!\cdots\!12$$$$p^{4} T^{15} +$$$$47\!\cdots\!04$$$$p^{8} T^{16} -$$$$16\!\cdots\!64$$$$p^{12} T^{17} +$$$$29\!\cdots\!96$$$$p^{16} T^{18} -$$$$70\!\cdots\!44$$$$p^{21} T^{19} +$$$$11\!\cdots\!69$$$$p^{24} T^{20} -$$$$19\!\cdots\!04$$$$p^{28} T^{21} +$$$$25\!\cdots\!72$$$$p^{32} T^{22} - 35007990661500345608 p^{36} T^{23} + 7013576890902382 p^{40} T^{24} - 1422808826512 p^{44} T^{25} + 209879072 p^{48} T^{26} - 20488 p^{52} T^{27} + p^{56} T^{28}$$
83 $$( 1 + 10112 T + 225278652 T^{2} + 1598383207850 T^{3} + 22827707456046842 T^{4} +$$$$13\!\cdots\!20$$$$T^{5} +$$$$15\!\cdots\!69$$$$T^{6} +$$$$80\!\cdots\!92$$$$T^{7} +$$$$15\!\cdots\!69$$$$p^{4} T^{8} +$$$$13\!\cdots\!20$$$$p^{8} T^{9} + 22827707456046842 p^{12} T^{10} + 1598383207850 p^{16} T^{11} + 225278652 p^{20} T^{12} + 10112 p^{24} T^{13} + p^{28} T^{14} )^{2}$$
89 $$1 - 13864 T + 96105248 T^{2} - 1285975306536 T^{3} + 12718246049793051 T^{4} - 31352269526314024464 T^{5} +$$$$39\!\cdots\!96$$$$T^{6} +$$$$11\!\cdots\!04$$$$T^{7} -$$$$42\!\cdots\!59$$$$T^{8} +$$$$26\!\cdots\!36$$$$T^{9} -$$$$10\!\cdots\!64$$$$T^{10} +$$$$52\!\cdots\!16$$$$T^{11} +$$$$20\!\cdots\!27$$$$T^{12} -$$$$26\!\cdots\!28$$$$p T^{13} +$$$$17\!\cdots\!88$$$$p^{2} T^{14} -$$$$26\!\cdots\!28$$$$p^{5} T^{15} +$$$$20\!\cdots\!27$$$$p^{8} T^{16} +$$$$52\!\cdots\!16$$$$p^{12} T^{17} -$$$$10\!\cdots\!64$$$$p^{16} T^{18} +$$$$26\!\cdots\!36$$$$p^{20} T^{19} -$$$$42\!\cdots\!59$$$$p^{24} T^{20} +$$$$11\!\cdots\!04$$$$p^{28} T^{21} +$$$$39\!\cdots\!96$$$$p^{32} T^{22} - 31352269526314024464 p^{36} T^{23} + 12718246049793051 p^{40} T^{24} - 1285975306536 p^{44} T^{25} + 96105248 p^{48} T^{26} - 13864 p^{52} T^{27} + p^{56} T^{28}$$
97 $$1 - 16622 T + 138145442 T^{2} - 1179500170998 T^{3} + 25150805825710411 T^{4} -$$$$37\!\cdots\!60$$$$T^{5} +$$$$34\!\cdots\!60$$$$T^{6} -$$$$34\!\cdots\!16$$$$T^{7} +$$$$40\!\cdots\!25$$$$T^{8} -$$$$44\!\cdots\!34$$$$T^{9} +$$$$42\!\cdots\!06$$$$T^{10} -$$$$47\!\cdots\!70$$$$T^{11} +$$$$53\!\cdots\!63$$$$T^{12} -$$$$44\!\cdots\!00$$$$T^{13} +$$$$37\!\cdots\!16$$$$T^{14} -$$$$44\!\cdots\!00$$$$p^{4} T^{15} +$$$$53\!\cdots\!63$$$$p^{8} T^{16} -$$$$47\!\cdots\!70$$$$p^{12} T^{17} +$$$$42\!\cdots\!06$$$$p^{16} T^{18} -$$$$44\!\cdots\!34$$$$p^{20} T^{19} +$$$$40\!\cdots\!25$$$$p^{24} T^{20} -$$$$34\!\cdots\!16$$$$p^{28} T^{21} +$$$$34\!\cdots\!60$$$$p^{32} T^{22} -$$$$37\!\cdots\!60$$$$p^{36} T^{23} + 25150805825710411 p^{40} T^{24} - 1179500170998 p^{44} T^{25} + 138145442 p^{48} T^{26} - 16622 p^{52} T^{27} + p^{56} T^{28}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{28} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$