# Properties

 Label 32-300e16-1.1-c8e16-0-0 Degree $32$ Conductor $4.305\times 10^{39}$ Sign $1$ Analytic cond. $2.47691\times 10^{33}$ Root an. cond. $11.0550$ Motivic weight $8$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 4.22e3·7-s + 2.36e4·11-s + 1.89e4·13-s + 4.49e4·17-s − 1.96e5·23-s + 3.74e6·31-s + 2.14e6·37-s + 1.63e7·41-s − 1.20e7·43-s + 1.49e7·47-s + 8.90e6·49-s − 2.37e7·53-s + 8.54e7·61-s + 9.94e7·67-s + 7.33e7·71-s − 1.24e8·73-s − 9.96e7·77-s − 1.91e7·81-s + 2.20e7·83-s − 7.97e7·91-s − 1.85e8·97-s − 9.67e7·101-s + 2.07e8·103-s − 2.98e8·107-s + 7.51e7·113-s − 1.89e8·119-s − 1.35e9·121-s + ⋯
 L(s)  = 1 − 1.75·7-s + 1.61·11-s + 0.661·13-s + 0.538·17-s − 0.701·23-s + 4.05·31-s + 1.14·37-s + 5.78·41-s − 3.53·43-s + 3.06·47-s + 1.54·49-s − 3.01·53-s + 6.16·61-s + 4.93·67-s + 2.88·71-s − 4.36·73-s − 2.83·77-s − 4/9·81-s + 0.464·83-s − 1.16·91-s − 2.09·97-s − 0.929·101-s + 1.84·103-s − 2.27·107-s + 0.460·113-s − 0.945·119-s − 6.31·121-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s+4)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$32$$ Conductor: $$2^{32} \cdot 3^{16} \cdot 5^{32}$$ Sign: $1$ Analytic conductor: $$2.47691\times 10^{33}$$ Root analytic conductor: $$11.0550$$ Motivic weight: $$8$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{300} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(32,\ 2^{32} \cdot 3^{16} \cdot 5^{32} ,\ ( \ : [4]^{16} ),\ 1 )$$

## Particular Values

 $$L(\frac{9}{2})$$ $$\approx$$ $$0.06747583437$$ $$L(\frac12)$$ $$\approx$$ $$0.06747583437$$ $$L(5)$$ 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 + p^{14} T^{4} )^{4}$$
5 $$1$$
good7 $$1 + 4220 T + 8904200 T^{2} + 20907912260 T^{3} - 20094734076324 T^{4} - 17913687169918700 p T^{5} - 2687191638797246600 p^{2} T^{6} -$$$$62\!\cdots\!20$$$$p^{3} T^{7} +$$$$10\!\cdots\!20$$$$p^{4} T^{8} +$$$$54\!\cdots\!20$$$$p^{5} T^{9} +$$$$14\!\cdots\!00$$$$p^{6} T^{10} +$$$$50\!\cdots\!00$$$$p^{7} T^{11} +$$$$78\!\cdots\!64$$$$p^{8} T^{12} +$$$$99\!\cdots\!60$$$$p^{9} T^{13} +$$$$65\!\cdots\!00$$$$p^{10} T^{14} +$$$$22\!\cdots\!00$$$$p^{11} T^{15} +$$$$69\!\cdots\!34$$$$p^{12} T^{16} +$$$$22\!\cdots\!00$$$$p^{19} T^{17} +$$$$65\!\cdots\!00$$$$p^{26} T^{18} +$$$$99\!\cdots\!60$$$$p^{33} T^{19} +$$$$78\!\cdots\!64$$$$p^{40} T^{20} +$$$$50\!\cdots\!00$$$$p^{47} T^{21} +$$$$14\!\cdots\!00$$$$p^{54} T^{22} +$$$$54\!\cdots\!20$$$$p^{61} T^{23} +$$$$10\!\cdots\!20$$$$p^{68} T^{24} -$$$$62\!\cdots\!20$$$$p^{75} T^{25} - 2687191638797246600 p^{82} T^{26} - 17913687169918700 p^{89} T^{27} - 20094734076324 p^{96} T^{28} + 20907912260 p^{104} T^{29} + 8904200 p^{112} T^{30} + 4220 p^{120} T^{31} + p^{128} T^{32}$$
11 $$( 1 - 11808 T + 885981526 T^{2} - 755524638672 p T^{3} + 27174596980964276 p T^{4} -$$$$19\!\cdots\!84$$$$T^{5} +$$$$44\!\cdots\!74$$$$T^{6} -$$$$14\!\cdots\!56$$$$T^{7} +$$$$46\!\cdots\!62$$$$T^{8} -$$$$14\!\cdots\!56$$$$p^{8} T^{9} +$$$$44\!\cdots\!74$$$$p^{16} T^{10} -$$$$19\!\cdots\!84$$$$p^{24} T^{11} + 27174596980964276 p^{33} T^{12} - 755524638672 p^{41} T^{13} + 885981526 p^{48} T^{14} - 11808 p^{56} T^{15} + p^{64} T^{16} )^{2}$$
13 $$1 - 18900 T + 178605000 T^{2} - 9923970627900 T^{3} - 118782183274202572 T^{4} +$$$$22\!\cdots\!00$$$$T^{5} -$$$$34\!\cdots\!00$$$$T^{6} +$$$$24\!\cdots\!00$$$$T^{7} -$$$$61\!\cdots\!32$$$$T^{8} +$$$$23\!\cdots\!00$$$$p T^{9} +$$$$11\!\cdots\!00$$$$T^{10} +$$$$64\!\cdots\!00$$$$T^{11} +$$$$10\!\cdots\!76$$$$T^{12} -$$$$77\!\cdots\!00$$$$T^{13} +$$$$22\!\cdots\!00$$$$T^{14} -$$$$50\!\cdots\!00$$$$T^{15} +$$$$22\!\cdots\!70$$$$T^{16} -$$$$50\!\cdots\!00$$$$p^{8} T^{17} +$$$$22\!\cdots\!00$$$$p^{16} T^{18} -$$$$77\!\cdots\!00$$$$p^{24} T^{19} +$$$$10\!\cdots\!76$$$$p^{32} T^{20} +$$$$64\!\cdots\!00$$$$p^{40} T^{21} +$$$$11\!\cdots\!00$$$$p^{48} T^{22} +$$$$23\!\cdots\!00$$$$p^{57} T^{23} -$$$$61\!\cdots\!32$$$$p^{64} T^{24} +$$$$24\!\cdots\!00$$$$p^{72} T^{25} -$$$$34\!\cdots\!00$$$$p^{80} T^{26} +$$$$22\!\cdots\!00$$$$p^{88} T^{27} - 118782183274202572 p^{96} T^{28} - 9923970627900 p^{104} T^{29} + 178605000 p^{112} T^{30} - 18900 p^{120} T^{31} + p^{128} T^{32}$$
17 $$1 - 44940 T + 1009801800 T^{2} + 153003810785580 T^{3} - 45000732490795208684 T^{4} +$$$$66\!\cdots\!00$$$$p^{2} T^{5} -$$$$29\!\cdots\!00$$$$T^{6} +$$$$69\!\cdots\!20$$$$T^{7} +$$$$12\!\cdots\!40$$$$T^{8} +$$$$37\!\cdots\!20$$$$T^{9} -$$$$13\!\cdots\!00$$$$T^{10} +$$$$13\!\cdots\!00$$$$T^{11} -$$$$92\!\cdots\!16$$$$T^{12} -$$$$28\!\cdots\!40$$$$T^{13} +$$$$50\!\cdots\!00$$$$T^{14} -$$$$27\!\cdots\!00$$$$T^{15} +$$$$61\!\cdots\!94$$$$T^{16} -$$$$27\!\cdots\!00$$$$p^{8} T^{17} +$$$$50\!\cdots\!00$$$$p^{16} T^{18} -$$$$28\!\cdots\!40$$$$p^{24} T^{19} -$$$$92\!\cdots\!16$$$$p^{32} T^{20} +$$$$13\!\cdots\!00$$$$p^{40} T^{21} -$$$$13\!\cdots\!00$$$$p^{48} T^{22} +$$$$37\!\cdots\!20$$$$p^{56} T^{23} +$$$$12\!\cdots\!40$$$$p^{64} T^{24} +$$$$69\!\cdots\!20$$$$p^{72} T^{25} -$$$$29\!\cdots\!00$$$$p^{80} T^{26} +$$$$66\!\cdots\!00$$$$p^{90} T^{27} - 45000732490795208684 p^{96} T^{28} + 153003810785580 p^{104} T^{29} + 1009801800 p^{112} T^{30} - 44940 p^{120} T^{31} + p^{128} T^{32}$$
19 $$1 - 196811814056 T^{2} +$$$$18\!\cdots\!20$$$$T^{4} -$$$$10\!\cdots\!60$$$$T^{6} +$$$$46\!\cdots\!20$$$$T^{8} -$$$$15\!\cdots\!68$$$$T^{10} +$$$$39\!\cdots\!28$$$$T^{12} -$$$$85\!\cdots\!40$$$$T^{14} +$$$$15\!\cdots\!70$$$$T^{16} -$$$$85\!\cdots\!40$$$$p^{16} T^{18} +$$$$39\!\cdots\!28$$$$p^{32} T^{20} -$$$$15\!\cdots\!68$$$$p^{48} T^{22} +$$$$46\!\cdots\!20$$$$p^{64} T^{24} -$$$$10\!\cdots\!60$$$$p^{80} T^{26} +$$$$18\!\cdots\!20$$$$p^{96} T^{28} - 196811814056 p^{112} T^{30} + p^{128} T^{32}$$
23 $$1 + 196440 T + 19294336800 T^{2} - 24795563077892040 T^{3} -$$$$84\!\cdots\!88$$$$T^{4} -$$$$82\!\cdots\!80$$$$T^{5} +$$$$31\!\cdots\!00$$$$T^{6} -$$$$32\!\cdots\!20$$$$T^{7} -$$$$26\!\cdots\!44$$$$p T^{8} -$$$$94\!\cdots\!60$$$$T^{9} +$$$$31\!\cdots\!00$$$$T^{10} +$$$$24\!\cdots\!60$$$$T^{11} +$$$$35\!\cdots\!64$$$$T^{12} -$$$$94\!\cdots\!00$$$$T^{13} -$$$$32\!\cdots\!00$$$$T^{14} +$$$$10\!\cdots\!00$$$$T^{15} +$$$$18\!\cdots\!70$$$$T^{16} +$$$$10\!\cdots\!00$$$$p^{8} T^{17} -$$$$32\!\cdots\!00$$$$p^{16} T^{18} -$$$$94\!\cdots\!00$$$$p^{24} T^{19} +$$$$35\!\cdots\!64$$$$p^{32} T^{20} +$$$$24\!\cdots\!60$$$$p^{40} T^{21} +$$$$31\!\cdots\!00$$$$p^{48} T^{22} -$$$$94\!\cdots\!60$$$$p^{56} T^{23} -$$$$26\!\cdots\!44$$$$p^{65} T^{24} -$$$$32\!\cdots\!20$$$$p^{72} T^{25} +$$$$31\!\cdots\!00$$$$p^{80} T^{26} -$$$$82\!\cdots\!80$$$$p^{88} T^{27} -$$$$84\!\cdots\!88$$$$p^{96} T^{28} - 24795563077892040 p^{104} T^{29} + 19294336800 p^{112} T^{30} + 196440 p^{120} T^{31} + p^{128} T^{32}$$
29 $$1 - 3378900968564 T^{2} +$$$$54\!\cdots\!60$$$$T^{4} -$$$$59\!\cdots\!80$$$$T^{6} +$$$$48\!\cdots\!60$$$$T^{8} -$$$$33\!\cdots\!72$$$$T^{10} +$$$$20\!\cdots\!08$$$$T^{12} -$$$$10\!\cdots\!80$$$$T^{14} +$$$$56\!\cdots\!90$$$$T^{16} -$$$$10\!\cdots\!80$$$$p^{16} T^{18} +$$$$20\!\cdots\!08$$$$p^{32} T^{20} -$$$$33\!\cdots\!72$$$$p^{48} T^{22} +$$$$48\!\cdots\!60$$$$p^{64} T^{24} -$$$$59\!\cdots\!80$$$$p^{80} T^{26} +$$$$54\!\cdots\!60$$$$p^{96} T^{28} - 3378900968564 p^{112} T^{30} + p^{128} T^{32}$$
31 $$( 1 - 1871312 T + 131590911936 p T^{2} - 4834324811235482608 T^{3} +$$$$71\!\cdots\!36$$$$T^{4} -$$$$22\!\cdots\!96$$$$p T^{5} +$$$$88\!\cdots\!84$$$$T^{6} -$$$$74\!\cdots\!04$$$$T^{7} +$$$$82\!\cdots\!62$$$$T^{8} -$$$$74\!\cdots\!04$$$$p^{8} T^{9} +$$$$88\!\cdots\!84$$$$p^{16} T^{10} -$$$$22\!\cdots\!96$$$$p^{25} T^{11} +$$$$71\!\cdots\!36$$$$p^{32} T^{12} - 4834324811235482608 p^{40} T^{13} + 131590911936 p^{49} T^{14} - 1871312 p^{56} T^{15} + p^{64} T^{16} )^{2}$$
37 $$1 - 2141100 T + 2292154605000 T^{2} + 9239158746994923900 T^{3} -$$$$28\!\cdots\!28$$$$T^{4} +$$$$21\!\cdots\!00$$$$T^{5} +$$$$62\!\cdots\!00$$$$T^{6} -$$$$15\!\cdots\!00$$$$T^{7} +$$$$75\!\cdots\!68$$$$T^{8} +$$$$42\!\cdots\!00$$$$T^{9} -$$$$27\!\cdots\!00$$$$T^{10} -$$$$16\!\cdots\!00$$$$T^{11} +$$$$36\!\cdots\!24$$$$T^{12} -$$$$33\!\cdots\!00$$$$p T^{13} -$$$$89\!\cdots\!00$$$$T^{14} +$$$$93\!\cdots\!00$$$$T^{15} -$$$$21\!\cdots\!30$$$$T^{16} +$$$$93\!\cdots\!00$$$$p^{8} T^{17} -$$$$89\!\cdots\!00$$$$p^{16} T^{18} -$$$$33\!\cdots\!00$$$$p^{25} T^{19} +$$$$36\!\cdots\!24$$$$p^{32} T^{20} -$$$$16\!\cdots\!00$$$$p^{40} T^{21} -$$$$27\!\cdots\!00$$$$p^{48} T^{22} +$$$$42\!\cdots\!00$$$$p^{56} T^{23} +$$$$75\!\cdots\!68$$$$p^{64} T^{24} -$$$$15\!\cdots\!00$$$$p^{72} T^{25} +$$$$62\!\cdots\!00$$$$p^{80} T^{26} +$$$$21\!\cdots\!00$$$$p^{88} T^{27} -$$$$28\!\cdots\!28$$$$p^{96} T^{28} + 9239158746994923900 p^{104} T^{29} + 2292154605000 p^{112} T^{30} - 2141100 p^{120} T^{31} + p^{128} T^{32}$$
41 $$( 1 - 8173500 T + 66110933231068 T^{2} -$$$$31\!\cdots\!00$$$$T^{3} +$$$$34\!\cdots\!28$$$$p T^{4} -$$$$45\!\cdots\!00$$$$T^{5} +$$$$14\!\cdots\!16$$$$T^{6} -$$$$38\!\cdots\!00$$$$T^{7} +$$$$11\!\cdots\!70$$$$T^{8} -$$$$38\!\cdots\!00$$$$p^{8} T^{9} +$$$$14\!\cdots\!16$$$$p^{16} T^{10} -$$$$45\!\cdots\!00$$$$p^{24} T^{11} +$$$$34\!\cdots\!28$$$$p^{33} T^{12} -$$$$31\!\cdots\!00$$$$p^{40} T^{13} + 66110933231068 p^{48} T^{14} - 8173500 p^{56} T^{15} + p^{64} T^{16} )^{2}$$
43 $$1 + 12080280 T + 72966582439200 T^{2} +$$$$29\!\cdots\!20$$$$T^{3} +$$$$65\!\cdots\!92$$$$T^{4} +$$$$40\!\cdots\!40$$$$T^{5} +$$$$62\!\cdots\!00$$$$T^{6} +$$$$29\!\cdots\!60$$$$T^{7} +$$$$29\!\cdots\!28$$$$T^{8} +$$$$14\!\cdots\!80$$$$T^{9} +$$$$44\!\cdots\!00$$$$T^{10} +$$$$84\!\cdots\!20$$$$T^{11} +$$$$65\!\cdots\!08$$$$p T^{12} +$$$$10\!\cdots\!00$$$$T^{13} +$$$$34\!\cdots\!00$$$$T^{14} +$$$$25\!\cdots\!00$$$$T^{15} +$$$$11\!\cdots\!70$$$$T^{16} +$$$$25\!\cdots\!00$$$$p^{8} T^{17} +$$$$34\!\cdots\!00$$$$p^{16} T^{18} +$$$$10\!\cdots\!00$$$$p^{24} T^{19} +$$$$65\!\cdots\!08$$$$p^{33} T^{20} +$$$$84\!\cdots\!20$$$$p^{40} T^{21} +$$$$44\!\cdots\!00$$$$p^{48} T^{22} +$$$$14\!\cdots\!80$$$$p^{56} T^{23} +$$$$29\!\cdots\!28$$$$p^{64} T^{24} +$$$$29\!\cdots\!60$$$$p^{72} T^{25} +$$$$62\!\cdots\!00$$$$p^{80} T^{26} +$$$$40\!\cdots\!40$$$$p^{88} T^{27} +$$$$65\!\cdots\!92$$$$p^{96} T^{28} +$$$$29\!\cdots\!20$$$$p^{104} T^{29} + 72966582439200 p^{112} T^{30} + 12080280 p^{120} T^{31} + p^{128} T^{32}$$
47 $$1 - 14942400 T + 111637658880000 T^{2} -$$$$94\!\cdots\!00$$$$T^{3} +$$$$81\!\cdots\!56$$$$T^{4} -$$$$48\!\cdots\!00$$$$T^{5} +$$$$26\!\cdots\!00$$$$T^{6} -$$$$17\!\cdots\!00$$$$T^{7} +$$$$90\!\cdots\!40$$$$T^{8} -$$$$35\!\cdots\!00$$$$T^{9} +$$$$16\!\cdots\!00$$$$T^{10} -$$$$82\!\cdots\!00$$$$T^{11} +$$$$18\!\cdots\!64$$$$T^{12} -$$$$20\!\cdots\!00$$$$T^{13} +$$$$21\!\cdots\!00$$$$T^{14} +$$$$31\!\cdots\!00$$$$T^{15} -$$$$97\!\cdots\!66$$$$T^{16} +$$$$31\!\cdots\!00$$$$p^{8} T^{17} +$$$$21\!\cdots\!00$$$$p^{16} T^{18} -$$$$20\!\cdots\!00$$$$p^{24} T^{19} +$$$$18\!\cdots\!64$$$$p^{32} T^{20} -$$$$82\!\cdots\!00$$$$p^{40} T^{21} +$$$$16\!\cdots\!00$$$$p^{48} T^{22} -$$$$35\!\cdots\!00$$$$p^{56} T^{23} +$$$$90\!\cdots\!40$$$$p^{64} T^{24} -$$$$17\!\cdots\!00$$$$p^{72} T^{25} +$$$$26\!\cdots\!00$$$$p^{80} T^{26} -$$$$48\!\cdots\!00$$$$p^{88} T^{27} +$$$$81\!\cdots\!56$$$$p^{96} T^{28} -$$$$94\!\cdots\!00$$$$p^{104} T^{29} + 111637658880000 p^{112} T^{30} - 14942400 p^{120} T^{31} + p^{128} T^{32}$$
53 $$1 + 23760300 T + 282275928045000 T^{2} +$$$$34\!\cdots\!00$$$$T^{3} +$$$$50\!\cdots\!76$$$$T^{4} +$$$$54\!\cdots\!00$$$$T^{5} +$$$$46\!\cdots\!00$$$$T^{6} +$$$$42\!\cdots\!00$$$$T^{7} +$$$$36\!\cdots\!80$$$$T^{8} +$$$$21\!\cdots\!00$$$$T^{9} +$$$$95\!\cdots\!00$$$$T^{10} +$$$$20\!\cdots\!00$$$$T^{11} -$$$$76\!\cdots\!16$$$$T^{12} -$$$$12\!\cdots\!00$$$$T^{13} -$$$$12\!\cdots\!00$$$$T^{14} -$$$$12\!\cdots\!00$$$$T^{15} -$$$$11\!\cdots\!06$$$$T^{16} -$$$$12\!\cdots\!00$$$$p^{8} T^{17} -$$$$12\!\cdots\!00$$$$p^{16} T^{18} -$$$$12\!\cdots\!00$$$$p^{24} T^{19} -$$$$76\!\cdots\!16$$$$p^{32} T^{20} +$$$$20\!\cdots\!00$$$$p^{40} T^{21} +$$$$95\!\cdots\!00$$$$p^{48} T^{22} +$$$$21\!\cdots\!00$$$$p^{56} T^{23} +$$$$36\!\cdots\!80$$$$p^{64} T^{24} +$$$$42\!\cdots\!00$$$$p^{72} T^{25} +$$$$46\!\cdots\!00$$$$p^{80} T^{26} +$$$$54\!\cdots\!00$$$$p^{88} T^{27} +$$$$50\!\cdots\!76$$$$p^{96} T^{28} +$$$$34\!\cdots\!00$$$$p^{104} T^{29} + 282275928045000 p^{112} T^{30} + 23760300 p^{120} T^{31} + p^{128} T^{32}$$
59 $$1 - 1591699600522124 T^{2} +$$$$12\!\cdots\!80$$$$T^{4} -$$$$61\!\cdots\!00$$$$T^{6} +$$$$22\!\cdots\!80$$$$T^{8} -$$$$61\!\cdots\!92$$$$T^{10} +$$$$14\!\cdots\!48$$$$T^{12} -$$$$26\!\cdots\!40$$$$T^{14} +$$$$42\!\cdots\!50$$$$T^{16} -$$$$26\!\cdots\!40$$$$p^{16} T^{18} +$$$$14\!\cdots\!48$$$$p^{32} T^{20} -$$$$61\!\cdots\!92$$$$p^{48} T^{22} +$$$$22\!\cdots\!80$$$$p^{64} T^{24} -$$$$61\!\cdots\!00$$$$p^{80} T^{26} +$$$$12\!\cdots\!80$$$$p^{96} T^{28} - 1591699600522124 p^{112} T^{30} + p^{128} T^{32}$$
61 $$( 1 - 42700956 T + 2023529928577820 T^{2} -$$$$55\!\cdots\!60$$$$T^{3} +$$$$15\!\cdots\!20$$$$T^{4} -$$$$30\!\cdots\!68$$$$T^{5} +$$$$61\!\cdots\!28$$$$T^{6} -$$$$97\!\cdots\!40$$$$T^{7} +$$$$15\!\cdots\!70$$$$T^{8} -$$$$97\!\cdots\!40$$$$p^{8} T^{9} +$$$$61\!\cdots\!28$$$$p^{16} T^{10} -$$$$30\!\cdots\!68$$$$p^{24} T^{11} +$$$$15\!\cdots\!20$$$$p^{32} T^{12} -$$$$55\!\cdots\!60$$$$p^{40} T^{13} + 2023529928577820 p^{48} T^{14} - 42700956 p^{56} T^{15} + p^{64} T^{16} )^{2}$$
67 $$1 - 99451240 T + 4945274568768800 T^{2} -$$$$16\!\cdots\!20$$$$T^{3} +$$$$45\!\cdots\!56$$$$T^{4} -$$$$10\!\cdots\!00$$$$T^{5} +$$$$17\!\cdots\!00$$$$T^{6} -$$$$19\!\cdots\!80$$$$T^{7} -$$$$13\!\cdots\!80$$$$T^{8} +$$$$13\!\cdots\!20$$$$T^{9} -$$$$40\!\cdots\!00$$$$T^{10} +$$$$86\!\cdots\!00$$$$T^{11} -$$$$13\!\cdots\!76$$$$T^{12} +$$$$12\!\cdots\!60$$$$T^{13} +$$$$92\!\cdots\!00$$$$T^{14} -$$$$74\!\cdots\!00$$$$T^{15} +$$$$19\!\cdots\!14$$$$T^{16} -$$$$74\!\cdots\!00$$$$p^{8} T^{17} +$$$$92\!\cdots\!00$$$$p^{16} T^{18} +$$$$12\!\cdots\!60$$$$p^{24} T^{19} -$$$$13\!\cdots\!76$$$$p^{32} T^{20} +$$$$86\!\cdots\!00$$$$p^{40} T^{21} -$$$$40\!\cdots\!00$$$$p^{48} T^{22} +$$$$13\!\cdots\!20$$$$p^{56} T^{23} -$$$$13\!\cdots\!80$$$$p^{64} T^{24} -$$$$19\!\cdots\!80$$$$p^{72} T^{25} +$$$$17\!\cdots\!00$$$$p^{80} T^{26} -$$$$10\!\cdots\!00$$$$p^{88} T^{27} +$$$$45\!\cdots\!56$$$$p^{96} T^{28} -$$$$16\!\cdots\!20$$$$p^{104} T^{29} + 4945274568768800 p^{112} T^{30} - 99451240 p^{120} T^{31} + p^{128} T^{32}$$
71 $$( 1 - 36651240 T + 3474509608622488 T^{2} -$$$$10\!\cdots\!80$$$$T^{3} +$$$$51\!\cdots\!88$$$$T^{4} -$$$$13\!\cdots\!40$$$$T^{5} +$$$$45\!\cdots\!36$$$$T^{6} -$$$$11\!\cdots\!00$$$$T^{7} +$$$$31\!\cdots\!70$$$$T^{8} -$$$$11\!\cdots\!00$$$$p^{8} T^{9} +$$$$45\!\cdots\!36$$$$p^{16} T^{10} -$$$$13\!\cdots\!40$$$$p^{24} T^{11} +$$$$51\!\cdots\!88$$$$p^{32} T^{12} -$$$$10\!\cdots\!80$$$$p^{40} T^{13} + 3474509608622488 p^{48} T^{14} - 36651240 p^{56} T^{15} + p^{64} T^{16} )^{2}$$
73 $$1 + 124097320 T + 7700072415591200 T^{2} +$$$$35\!\cdots\!60$$$$T^{3} +$$$$14\!\cdots\!76$$$$T^{4} +$$$$46\!\cdots\!00$$$$T^{5} +$$$$12\!\cdots\!00$$$$T^{6} +$$$$24\!\cdots\!40$$$$T^{7} +$$$$14\!\cdots\!00$$$$T^{8} -$$$$13\!\cdots\!60$$$$T^{9} -$$$$76\!\cdots\!00$$$$T^{10} -$$$$26\!\cdots\!00$$$$T^{11} -$$$$64\!\cdots\!76$$$$T^{12} -$$$$96\!\cdots\!80$$$$T^{13} +$$$$27\!\cdots\!00$$$$T^{14} +$$$$82\!\cdots\!00$$$$T^{15} +$$$$32\!\cdots\!14$$$$T^{16} +$$$$82\!\cdots\!00$$$$p^{8} T^{17} +$$$$27\!\cdots\!00$$$$p^{16} T^{18} -$$$$96\!\cdots\!80$$$$p^{24} T^{19} -$$$$64\!\cdots\!76$$$$p^{32} T^{20} -$$$$26\!\cdots\!00$$$$p^{40} T^{21} -$$$$76\!\cdots\!00$$$$p^{48} T^{22} -$$$$13\!\cdots\!60$$$$p^{56} T^{23} +$$$$14\!\cdots\!00$$$$p^{64} T^{24} +$$$$24\!\cdots\!40$$$$p^{72} T^{25} +$$$$12\!\cdots\!00$$$$p^{80} T^{26} +$$$$46\!\cdots\!00$$$$p^{88} T^{27} +$$$$14\!\cdots\!76$$$$p^{96} T^{28} +$$$$35\!\cdots\!60$$$$p^{104} T^{29} + 7700072415591200 p^{112} T^{30} + 124097320 p^{120} T^{31} + p^{128} T^{32}$$
79 $$1 - 4824778998527008 T^{2} +$$$$12\!\cdots\!96$$$$T^{4} -$$$$22\!\cdots\!32$$$$T^{6} +$$$$32\!\cdots\!96$$$$T^{8} -$$$$39\!\cdots\!24$$$$T^{10} +$$$$61\!\cdots\!44$$$$T^{12} -$$$$11\!\cdots\!76$$$$T^{14} +$$$$19\!\cdots\!02$$$$T^{16} -$$$$11\!\cdots\!76$$$$p^{16} T^{18} +$$$$61\!\cdots\!44$$$$p^{32} T^{20} -$$$$39\!\cdots\!24$$$$p^{48} T^{22} +$$$$32\!\cdots\!96$$$$p^{64} T^{24} -$$$$22\!\cdots\!32$$$$p^{80} T^{26} +$$$$12\!\cdots\!96$$$$p^{96} T^{28} - 4824778998527008 p^{112} T^{30} + p^{128} T^{32}$$
83 $$1 - 22058160 T + 243281211292800 T^{2} -$$$$23\!\cdots\!40$$$$T^{3} +$$$$16\!\cdots\!52$$$$T^{4} +$$$$35\!\cdots\!20$$$$T^{5} +$$$$20\!\cdots\!00$$$$T^{6} +$$$$67\!\cdots\!80$$$$T^{7} -$$$$31\!\cdots\!92$$$$T^{8} -$$$$40\!\cdots\!60$$$$T^{9} -$$$$65\!\cdots\!00$$$$T^{10} -$$$$56\!\cdots\!40$$$$T^{11} +$$$$58\!\cdots\!04$$$$T^{12} +$$$$97\!\cdots\!00$$$$T^{13} +$$$$45\!\cdots\!00$$$$T^{14} -$$$$21\!\cdots\!00$$$$T^{15} -$$$$29\!\cdots\!30$$$$T^{16} -$$$$21\!\cdots\!00$$$$p^{8} T^{17} +$$$$45\!\cdots\!00$$$$p^{16} T^{18} +$$$$97\!\cdots\!00$$$$p^{24} T^{19} +$$$$58\!\cdots\!04$$$$p^{32} T^{20} -$$$$56\!\cdots\!40$$$$p^{40} T^{21} -$$$$65\!\cdots\!00$$$$p^{48} T^{22} -$$$$40\!\cdots\!60$$$$p^{56} T^{23} -$$$$31\!\cdots\!92$$$$p^{64} T^{24} +$$$$67\!\cdots\!80$$$$p^{72} T^{25} +$$$$20\!\cdots\!00$$$$p^{80} T^{26} +$$$$35\!\cdots\!20$$$$p^{88} T^{27} +$$$$16\!\cdots\!52$$$$p^{96} T^{28} -$$$$23\!\cdots\!40$$$$p^{104} T^{29} + 243281211292800 p^{112} T^{30} - 22058160 p^{120} T^{31} + p^{128} T^{32}$$
89 $$1 - 24631134882368696 T^{2} +$$$$34\!\cdots\!20$$$$T^{4} -$$$$34\!\cdots\!60$$$$T^{6} +$$$$26\!\cdots\!20$$$$T^{8} -$$$$17\!\cdots\!68$$$$T^{10} +$$$$95\!\cdots\!48$$$$T^{12} -$$$$45\!\cdots\!40$$$$T^{14} +$$$$19\!\cdots\!70$$$$T^{16} -$$$$45\!\cdots\!40$$$$p^{16} T^{18} +$$$$95\!\cdots\!48$$$$p^{32} T^{20} -$$$$17\!\cdots\!68$$$$p^{48} T^{22} +$$$$26\!\cdots\!20$$$$p^{64} T^{24} -$$$$34\!\cdots\!60$$$$p^{80} T^{26} +$$$$34\!\cdots\!20$$$$p^{96} T^{28} - 24631134882368696 p^{112} T^{30} + p^{128} T^{32}$$
97 $$1 + 185269800 T + 17162449396020000 T^{2} +$$$$26\!\cdots\!00$$$$T^{3} +$$$$23\!\cdots\!32$$$$T^{4} -$$$$15\!\cdots\!00$$$$T^{5} -$$$$74\!\cdots\!00$$$$T^{6} -$$$$17\!\cdots\!00$$$$T^{7} -$$$$42\!\cdots\!52$$$$T^{8} -$$$$37\!\cdots\!00$$$$T^{9} -$$$$24\!\cdots\!00$$$$T^{10} -$$$$28\!\cdots\!00$$$$T^{11} -$$$$95\!\cdots\!16$$$$T^{12} +$$$$14\!\cdots\!00$$$$T^{13} +$$$$14\!\cdots\!00$$$$T^{14} +$$$$21\!\cdots\!00$$$$T^{15} +$$$$30\!\cdots\!70$$$$T^{16} +$$$$21\!\cdots\!00$$$$p^{8} T^{17} +$$$$14\!\cdots\!00$$$$p^{16} T^{18} +$$$$14\!\cdots\!00$$$$p^{24} T^{19} -$$$$95\!\cdots\!16$$$$p^{32} T^{20} -$$$$28\!\cdots\!00$$$$p^{40} T^{21} -$$$$24\!\cdots\!00$$$$p^{48} T^{22} -$$$$37\!\cdots\!00$$$$p^{56} T^{23} -$$$$42\!\cdots\!52$$$$p^{64} T^{24} -$$$$17\!\cdots\!00$$$$p^{72} T^{25} -$$$$74\!\cdots\!00$$$$p^{80} T^{26} -$$$$15\!\cdots\!00$$$$p^{88} T^{27} +$$$$23\!\cdots\!32$$$$p^{96} T^{28} +$$$$26\!\cdots\!00$$$$p^{104} T^{29} + 17162449396020000 p^{112} T^{30} + 185269800 p^{120} T^{31} + p^{128} T^{32}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$