# Properties

 Label 32-273e16-1.1-c11e16-0-0 Degree $32$ Conductor $9.519\times 10^{38}$ Sign $1$ Analytic cond. $1.40438\times 10^{37}$ Root an. cond. $14.4830$ Motivic weight $11$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $16$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 63·2-s − 3.88e3·3-s − 4.01e3·4-s − 3.16e3·5-s + 2.44e5·6-s + 2.68e5·7-s + 3.16e5·8-s + 8.03e6·9-s + 1.99e5·10-s − 4.66e5·11-s + 1.56e7·12-s − 5.94e6·13-s − 1.69e7·14-s + 1.23e7·15-s + 7.38e6·16-s + 1.75e6·17-s − 5.05e8·18-s + 4.23e6·19-s + 1.27e7·20-s − 1.04e9·21-s + 2.94e7·22-s − 1.50e8·23-s − 1.23e9·24-s − 3.10e8·25-s + 3.74e8·26-s − 1.17e10·27-s − 1.08e9·28-s + ⋯
 L(s)  = 1 − 1.39·2-s − 9.23·3-s − 1.96·4-s − 0.453·5-s + 12.8·6-s + 6.04·7-s + 3.41·8-s + 45.3·9-s + 0.631·10-s − 0.874·11-s + 18.1·12-s − 4.43·13-s − 8.41·14-s + 4.18·15-s + 1.76·16-s + 0.299·17-s − 63.1·18-s + 0.392·19-s + 0.889·20-s − 55.8·21-s + 1.21·22-s − 4.87·23-s − 31.5·24-s − 6.35·25-s + 6.17·26-s − 157.·27-s − 11.8·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(12-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+11/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$32$$ Conductor: $$3^{16} \cdot 7^{16} \cdot 13^{16}$$ Sign: $1$ Analytic conductor: $$1.40438\times 10^{37}$$ Root analytic conductor: $$14.4830$$ Motivic weight: $$11$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{273} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$16$$ Selberg data: $$(32,\ 3^{16} \cdot 7^{16} \cdot 13^{16} ,\ ( \ : [11/2]^{16} ),\ 1 )$$

## Particular Values

 $$L(6)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{13}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$( 1 + p^{5} T )^{16}$$
7 $$( 1 - p^{5} T )^{16}$$
13 $$( 1 + p^{5} T )^{16}$$
good2 $$1 + 63 T + 1997 p^{2} T^{2} + 219771 p T^{3} + 4055519 p^{3} T^{4} + 6384663 p^{8} T^{5} + 3037487649 p^{5} T^{6} + 144569617173 p^{5} T^{7} + 482873893431 p^{9} T^{8} + 41210051108841 p^{8} T^{9} + 60521892083273 p^{13} T^{10} + 2239031303654031 p^{13} T^{11} + 11791744845855377 p^{16} T^{12} + 23806191912815115 p^{20} T^{13} + 68891401551621091 p^{24} T^{14} + 1023283383416203185 p^{25} T^{15} + 7892242921598689875 p^{28} T^{16} + 1023283383416203185 p^{36} T^{17} + 68891401551621091 p^{46} T^{18} + 23806191912815115 p^{53} T^{19} + 11791744845855377 p^{60} T^{20} + 2239031303654031 p^{68} T^{21} + 60521892083273 p^{79} T^{22} + 41210051108841 p^{85} T^{23} + 482873893431 p^{97} T^{24} + 144569617173 p^{104} T^{25} + 3037487649 p^{115} T^{26} + 6384663 p^{129} T^{27} + 4055519 p^{135} T^{28} + 219771 p^{144} T^{29} + 1997 p^{156} T^{30} + 63 p^{165} T^{31} + p^{176} T^{32}$$
5 $$1 + 3168 T + 320151524 T^{2} + 1023097332312 T^{3} + 52139051805256326 T^{4} + 32871579807780458208 p T^{5} +$$$$22\!\cdots\!56$$$$p^{2} T^{6} +$$$$14\!\cdots\!92$$$$p^{3} T^{7} +$$$$75\!\cdots\!21$$$$p^{4} T^{8} +$$$$99\!\cdots\!64$$$$p^{6} T^{9} +$$$$79\!\cdots\!68$$$$p^{8} T^{10} +$$$$14\!\cdots\!32$$$$p^{7} T^{11} +$$$$44\!\cdots\!18$$$$p^{8} T^{12} +$$$$35\!\cdots\!68$$$$p^{9} T^{13} +$$$$90\!\cdots\!04$$$$p^{10} T^{14} +$$$$15\!\cdots\!84$$$$p^{12} T^{15} +$$$$70\!\cdots\!24$$$$p^{14} T^{16} +$$$$15\!\cdots\!84$$$$p^{23} T^{17} +$$$$90\!\cdots\!04$$$$p^{32} T^{18} +$$$$35\!\cdots\!68$$$$p^{42} T^{19} +$$$$44\!\cdots\!18$$$$p^{52} T^{20} +$$$$14\!\cdots\!32$$$$p^{62} T^{21} +$$$$79\!\cdots\!68$$$$p^{74} T^{22} +$$$$99\!\cdots\!64$$$$p^{83} T^{23} +$$$$75\!\cdots\!21$$$$p^{92} T^{24} +$$$$14\!\cdots\!92$$$$p^{102} T^{25} +$$$$22\!\cdots\!56$$$$p^{112} T^{26} + 32871579807780458208 p^{122} T^{27} + 52139051805256326 p^{132} T^{28} + 1023097332312 p^{143} T^{29} + 320151524 p^{154} T^{30} + 3168 p^{165} T^{31} + p^{176} T^{32}$$
11 $$1 + 42444 p T + 1819831942074 T^{2} + 77178495131611128 p T^{3} +$$$$16\!\cdots\!05$$$$T^{4} +$$$$67\!\cdots\!24$$$$p T^{5} +$$$$10\!\cdots\!70$$$$T^{6} +$$$$38\!\cdots\!84$$$$p T^{7} +$$$$52\!\cdots\!10$$$$T^{8} +$$$$16\!\cdots\!32$$$$p T^{9} +$$$$21\!\cdots\!58$$$$T^{10} +$$$$64\!\cdots\!12$$$$p T^{11} +$$$$80\!\cdots\!23$$$$T^{12} +$$$$22\!\cdots\!12$$$$p T^{13} +$$$$26\!\cdots\!14$$$$T^{14} +$$$$70\!\cdots\!52$$$$p T^{15} +$$$$80\!\cdots\!94$$$$T^{16} +$$$$70\!\cdots\!52$$$$p^{12} T^{17} +$$$$26\!\cdots\!14$$$$p^{22} T^{18} +$$$$22\!\cdots\!12$$$$p^{34} T^{19} +$$$$80\!\cdots\!23$$$$p^{44} T^{20} +$$$$64\!\cdots\!12$$$$p^{56} T^{21} +$$$$21\!\cdots\!58$$$$p^{66} T^{22} +$$$$16\!\cdots\!32$$$$p^{78} T^{23} +$$$$52\!\cdots\!10$$$$p^{88} T^{24} +$$$$38\!\cdots\!84$$$$p^{100} T^{25} +$$$$10\!\cdots\!70$$$$p^{110} T^{26} +$$$$67\!\cdots\!24$$$$p^{122} T^{27} +$$$$16\!\cdots\!05$$$$p^{132} T^{28} + 77178495131611128 p^{144} T^{29} + 1819831942074 p^{154} T^{30} + 42444 p^{166} T^{31} + p^{176} T^{32}$$
17 $$1 - 1753452 T + 272778464406782 T^{2} -$$$$13\!\cdots\!20$$$$T^{3} +$$$$22\!\cdots\!53$$$$p T^{4} +$$$$25\!\cdots\!00$$$$T^{5} +$$$$20\!\cdots\!38$$$$p T^{6} +$$$$64\!\cdots\!80$$$$T^{7} +$$$$24\!\cdots\!78$$$$T^{8} +$$$$72\!\cdots\!00$$$$T^{9} +$$$$13\!\cdots\!58$$$$T^{10} +$$$$54\!\cdots\!16$$$$T^{11} +$$$$36\!\cdots\!55$$$$p T^{12} +$$$$30\!\cdots\!20$$$$T^{13} +$$$$87\!\cdots\!58$$$$p^{2} T^{14} +$$$$13\!\cdots\!80$$$$T^{15} +$$$$91\!\cdots\!42$$$$T^{16} +$$$$13\!\cdots\!80$$$$p^{11} T^{17} +$$$$87\!\cdots\!58$$$$p^{24} T^{18} +$$$$30\!\cdots\!20$$$$p^{33} T^{19} +$$$$36\!\cdots\!55$$$$p^{45} T^{20} +$$$$54\!\cdots\!16$$$$p^{55} T^{21} +$$$$13\!\cdots\!58$$$$p^{66} T^{22} +$$$$72\!\cdots\!00$$$$p^{77} T^{23} +$$$$24\!\cdots\!78$$$$p^{88} T^{24} +$$$$64\!\cdots\!80$$$$p^{99} T^{25} +$$$$20\!\cdots\!38$$$$p^{111} T^{26} +$$$$25\!\cdots\!00$$$$p^{121} T^{27} +$$$$22\!\cdots\!53$$$$p^{133} T^{28} -$$$$13\!\cdots\!20$$$$p^{143} T^{29} + 272778464406782 p^{154} T^{30} - 1753452 p^{165} T^{31} + p^{176} T^{32}$$
19 $$1 - 4237800 T + 1119294184111812 T^{2} -$$$$59\!\cdots\!80$$$$T^{3} +$$$$63\!\cdots\!94$$$$T^{4} -$$$$38\!\cdots\!04$$$$T^{5} +$$$$24\!\cdots\!36$$$$T^{6} -$$$$15\!\cdots\!28$$$$T^{7} +$$$$69\!\cdots\!69$$$$T^{8} -$$$$46\!\cdots\!92$$$$T^{9} +$$$$15\!\cdots\!00$$$$T^{10} -$$$$10\!\cdots\!48$$$$T^{11} +$$$$28\!\cdots\!86$$$$T^{12} -$$$$18\!\cdots\!12$$$$T^{13} +$$$$43\!\cdots\!92$$$$T^{14} -$$$$26\!\cdots\!80$$$$T^{15} +$$$$55\!\cdots\!28$$$$T^{16} -$$$$26\!\cdots\!80$$$$p^{11} T^{17} +$$$$43\!\cdots\!92$$$$p^{22} T^{18} -$$$$18\!\cdots\!12$$$$p^{33} T^{19} +$$$$28\!\cdots\!86$$$$p^{44} T^{20} -$$$$10\!\cdots\!48$$$$p^{55} T^{21} +$$$$15\!\cdots\!00$$$$p^{66} T^{22} -$$$$46\!\cdots\!92$$$$p^{77} T^{23} +$$$$69\!\cdots\!69$$$$p^{88} T^{24} -$$$$15\!\cdots\!28$$$$p^{99} T^{25} +$$$$24\!\cdots\!36$$$$p^{110} T^{26} -$$$$38\!\cdots\!04$$$$p^{121} T^{27} +$$$$63\!\cdots\!94$$$$p^{132} T^{28} -$$$$59\!\cdots\!80$$$$p^{143} T^{29} + 1119294184111812 p^{154} T^{30} - 4237800 p^{165} T^{31} + p^{176} T^{32}$$
23 $$1 + 150481440 T + 18318396308951624 T^{2} +$$$$15\!\cdots\!96$$$$T^{3} +$$$$10\!\cdots\!34$$$$T^{4} +$$$$64\!\cdots\!96$$$$T^{5} +$$$$34\!\cdots\!84$$$$T^{6} +$$$$16\!\cdots\!00$$$$T^{7} +$$$$70\!\cdots\!17$$$$T^{8} +$$$$27\!\cdots\!76$$$$T^{9} +$$$$10\!\cdots\!84$$$$T^{10} +$$$$33\!\cdots\!28$$$$T^{11} +$$$$11\!\cdots\!58$$$$T^{12} +$$$$33\!\cdots\!16$$$$T^{13} +$$$$10\!\cdots\!52$$$$T^{14} +$$$$30\!\cdots\!72$$$$T^{15} +$$$$93\!\cdots\!00$$$$T^{16} +$$$$30\!\cdots\!72$$$$p^{11} T^{17} +$$$$10\!\cdots\!52$$$$p^{22} T^{18} +$$$$33\!\cdots\!16$$$$p^{33} T^{19} +$$$$11\!\cdots\!58$$$$p^{44} T^{20} +$$$$33\!\cdots\!28$$$$p^{55} T^{21} +$$$$10\!\cdots\!84$$$$p^{66} T^{22} +$$$$27\!\cdots\!76$$$$p^{77} T^{23} +$$$$70\!\cdots\!17$$$$p^{88} T^{24} +$$$$16\!\cdots\!00$$$$p^{99} T^{25} +$$$$34\!\cdots\!84$$$$p^{110} T^{26} +$$$$64\!\cdots\!96$$$$p^{121} T^{27} +$$$$10\!\cdots\!34$$$$p^{132} T^{28} +$$$$15\!\cdots\!96$$$$p^{143} T^{29} + 18318396308951624 p^{154} T^{30} + 150481440 p^{165} T^{31} + p^{176} T^{32}$$
29 $$1 + 111411432 T + 140907589341449384 T^{2} +$$$$12\!\cdots\!04$$$$T^{3} +$$$$93\!\cdots\!98$$$$T^{4} +$$$$65\!\cdots\!08$$$$T^{5} +$$$$39\!\cdots\!28$$$$T^{6} +$$$$20\!\cdots\!36$$$$T^{7} +$$$$11\!\cdots\!25$$$$T^{8} +$$$$42\!\cdots\!12$$$$T^{9} +$$$$26\!\cdots\!56$$$$T^{10} +$$$$60\!\cdots\!92$$$$T^{11} +$$$$48\!\cdots\!70$$$$T^{12} +$$$$57\!\cdots\!04$$$$T^{13} +$$$$74\!\cdots\!64$$$$T^{14} +$$$$44\!\cdots\!24$$$$T^{15} +$$$$97\!\cdots\!96$$$$T^{16} +$$$$44\!\cdots\!24$$$$p^{11} T^{17} +$$$$74\!\cdots\!64$$$$p^{22} T^{18} +$$$$57\!\cdots\!04$$$$p^{33} T^{19} +$$$$48\!\cdots\!70$$$$p^{44} T^{20} +$$$$60\!\cdots\!92$$$$p^{55} T^{21} +$$$$26\!\cdots\!56$$$$p^{66} T^{22} +$$$$42\!\cdots\!12$$$$p^{77} T^{23} +$$$$11\!\cdots\!25$$$$p^{88} T^{24} +$$$$20\!\cdots\!36$$$$p^{99} T^{25} +$$$$39\!\cdots\!28$$$$p^{110} T^{26} +$$$$65\!\cdots\!08$$$$p^{121} T^{27} +$$$$93\!\cdots\!98$$$$p^{132} T^{28} +$$$$12\!\cdots\!04$$$$p^{143} T^{29} + 140907589341449384 p^{154} T^{30} + 111411432 p^{165} T^{31} + p^{176} T^{32}$$
31 $$1 + 90536236 T + 170987389409218902 T^{2} +$$$$94\!\cdots\!36$$$$T^{3} +$$$$14\!\cdots\!17$$$$T^{4} +$$$$42\!\cdots\!12$$$$T^{5} +$$$$78\!\cdots\!54$$$$T^{6} +$$$$85\!\cdots\!88$$$$T^{7} +$$$$33\!\cdots\!66$$$$T^{8} +$$$$12\!\cdots\!00$$$$T^{9} +$$$$11\!\cdots\!38$$$$T^{10} -$$$$28\!\cdots\!44$$$$T^{11} +$$$$37\!\cdots\!03$$$$T^{12} +$$$$22\!\cdots\!88$$$$T^{13} +$$$$10\!\cdots\!26$$$$T^{14} +$$$$13\!\cdots\!24$$$$T^{15} +$$$$28\!\cdots\!26$$$$T^{16} +$$$$13\!\cdots\!24$$$$p^{11} T^{17} +$$$$10\!\cdots\!26$$$$p^{22} T^{18} +$$$$22\!\cdots\!88$$$$p^{33} T^{19} +$$$$37\!\cdots\!03$$$$p^{44} T^{20} -$$$$28\!\cdots\!44$$$$p^{55} T^{21} +$$$$11\!\cdots\!38$$$$p^{66} T^{22} +$$$$12\!\cdots\!00$$$$p^{77} T^{23} +$$$$33\!\cdots\!66$$$$p^{88} T^{24} +$$$$85\!\cdots\!88$$$$p^{99} T^{25} +$$$$78\!\cdots\!54$$$$p^{110} T^{26} +$$$$42\!\cdots\!12$$$$p^{121} T^{27} +$$$$14\!\cdots\!17$$$$p^{132} T^{28} +$$$$94\!\cdots\!36$$$$p^{143} T^{29} + 170987389409218902 p^{154} T^{30} + 90536236 p^{165} T^{31} + p^{176} T^{32}$$
37 $$1 + 1756337900 T + 3428636590713366786 T^{2} +$$$$41\!\cdots\!56$$$$T^{3} +$$$$49\!\cdots\!53$$$$T^{4} +$$$$47\!\cdots\!92$$$$T^{5} +$$$$42\!\cdots\!70$$$$T^{6} +$$$$33\!\cdots\!36$$$$T^{7} +$$$$25\!\cdots\!90$$$$T^{8} +$$$$16\!\cdots\!04$$$$T^{9} +$$$$10\!\cdots\!10$$$$T^{10} +$$$$63\!\cdots\!92$$$$T^{11} +$$$$35\!\cdots\!95$$$$T^{12} +$$$$18\!\cdots\!44$$$$T^{13} +$$$$88\!\cdots\!46$$$$T^{14} +$$$$11\!\cdots\!88$$$$p T^{15} +$$$$17\!\cdots\!82$$$$T^{16} +$$$$11\!\cdots\!88$$$$p^{12} T^{17} +$$$$88\!\cdots\!46$$$$p^{22} T^{18} +$$$$18\!\cdots\!44$$$$p^{33} T^{19} +$$$$35\!\cdots\!95$$$$p^{44} T^{20} +$$$$63\!\cdots\!92$$$$p^{55} T^{21} +$$$$10\!\cdots\!10$$$$p^{66} T^{22} +$$$$16\!\cdots\!04$$$$p^{77} T^{23} +$$$$25\!\cdots\!90$$$$p^{88} T^{24} +$$$$33\!\cdots\!36$$$$p^{99} T^{25} +$$$$42\!\cdots\!70$$$$p^{110} T^{26} +$$$$47\!\cdots\!92$$$$p^{121} T^{27} +$$$$49\!\cdots\!53$$$$p^{132} T^{28} +$$$$41\!\cdots\!56$$$$p^{143} T^{29} + 3428636590713366786 p^{154} T^{30} + 1756337900 p^{165} T^{31} + p^{176} T^{32}$$
41 $$1 - 649575720 T + 5412000906547565728 T^{2} -$$$$36\!\cdots\!40$$$$T^{3} +$$$$14\!\cdots\!96$$$$T^{4} -$$$$10\!\cdots\!36$$$$T^{5} +$$$$26\!\cdots\!24$$$$T^{6} -$$$$18\!\cdots\!32$$$$T^{7} +$$$$35\!\cdots\!52$$$$T^{8} -$$$$23\!\cdots\!40$$$$T^{9} +$$$$37\!\cdots\!56$$$$T^{10} -$$$$23\!\cdots\!44$$$$T^{11} +$$$$32\!\cdots\!84$$$$T^{12} -$$$$19\!\cdots\!20$$$$T^{13} +$$$$22\!\cdots\!52$$$$T^{14} -$$$$12\!\cdots\!28$$$$T^{15} +$$$$13\!\cdots\!34$$$$T^{16} -$$$$12\!\cdots\!28$$$$p^{11} T^{17} +$$$$22\!\cdots\!52$$$$p^{22} T^{18} -$$$$19\!\cdots\!20$$$$p^{33} T^{19} +$$$$32\!\cdots\!84$$$$p^{44} T^{20} -$$$$23\!\cdots\!44$$$$p^{55} T^{21} +$$$$37\!\cdots\!56$$$$p^{66} T^{22} -$$$$23\!\cdots\!40$$$$p^{77} T^{23} +$$$$35\!\cdots\!52$$$$p^{88} T^{24} -$$$$18\!\cdots\!32$$$$p^{99} T^{25} +$$$$26\!\cdots\!24$$$$p^{110} T^{26} -$$$$10\!\cdots\!36$$$$p^{121} T^{27} +$$$$14\!\cdots\!96$$$$p^{132} T^{28} -$$$$36\!\cdots\!40$$$$p^{143} T^{29} + 5412000906547565728 p^{154} T^{30} - 649575720 p^{165} T^{31} + p^{176} T^{32}$$
43 $$1 + 1889554520 T + 10463376342718153752 T^{2} +$$$$16\!\cdots\!68$$$$T^{3} +$$$$51\!\cdots\!62$$$$T^{4} +$$$$72\!\cdots\!92$$$$T^{5} +$$$$15\!\cdots\!08$$$$T^{6} +$$$$19\!\cdots\!80$$$$T^{7} +$$$$35\!\cdots\!41$$$$T^{8} +$$$$39\!\cdots\!36$$$$T^{9} +$$$$59\!\cdots\!92$$$$T^{10} +$$$$61\!\cdots\!56$$$$T^{11} +$$$$81\!\cdots\!66$$$$T^{12} +$$$$77\!\cdots\!52$$$$T^{13} +$$$$94\!\cdots\!20$$$$T^{14} +$$$$83\!\cdots\!16$$$$T^{15} +$$$$21\!\cdots\!60$$$$p T^{16} +$$$$83\!\cdots\!16$$$$p^{11} T^{17} +$$$$94\!\cdots\!20$$$$p^{22} T^{18} +$$$$77\!\cdots\!52$$$$p^{33} T^{19} +$$$$81\!\cdots\!66$$$$p^{44} T^{20} +$$$$61\!\cdots\!56$$$$p^{55} T^{21} +$$$$59\!\cdots\!92$$$$p^{66} T^{22} +$$$$39\!\cdots\!36$$$$p^{77} T^{23} +$$$$35\!\cdots\!41$$$$p^{88} T^{24} +$$$$19\!\cdots\!80$$$$p^{99} T^{25} +$$$$15\!\cdots\!08$$$$p^{110} T^{26} +$$$$72\!\cdots\!92$$$$p^{121} T^{27} +$$$$51\!\cdots\!62$$$$p^{132} T^{28} +$$$$16\!\cdots\!68$$$$p^{143} T^{29} + 10463376342718153752 p^{154} T^{30} + 1889554520 p^{165} T^{31} + p^{176} T^{32}$$
47 $$1 + 4036082940 T + 27322802044134664402 T^{2} +$$$$77\!\cdots\!88$$$$T^{3} +$$$$30\!\cdots\!25$$$$T^{4} +$$$$67\!\cdots\!36$$$$T^{5} +$$$$20\!\cdots\!46$$$$T^{6} +$$$$34\!\cdots\!52$$$$T^{7} +$$$$88\!\cdots\!30$$$$T^{8} +$$$$11\!\cdots\!52$$$$T^{9} +$$$$27\!\cdots\!10$$$$T^{10} +$$$$24\!\cdots\!76$$$$T^{11} +$$$$61\!\cdots\!99$$$$T^{12} +$$$$26\!\cdots\!12$$$$T^{13} +$$$$11\!\cdots\!34$$$$T^{14} -$$$$10\!\cdots\!40$$$$T^{15} +$$$$23\!\cdots\!34$$$$T^{16} -$$$$10\!\cdots\!40$$$$p^{11} T^{17} +$$$$11\!\cdots\!34$$$$p^{22} T^{18} +$$$$26\!\cdots\!12$$$$p^{33} T^{19} +$$$$61\!\cdots\!99$$$$p^{44} T^{20} +$$$$24\!\cdots\!76$$$$p^{55} T^{21} +$$$$27\!\cdots\!10$$$$p^{66} T^{22} +$$$$11\!\cdots\!52$$$$p^{77} T^{23} +$$$$88\!\cdots\!30$$$$p^{88} T^{24} +$$$$34\!\cdots\!52$$$$p^{99} T^{25} +$$$$20\!\cdots\!46$$$$p^{110} T^{26} +$$$$67\!\cdots\!36$$$$p^{121} T^{27} +$$$$30\!\cdots\!25$$$$p^{132} T^{28} +$$$$77\!\cdots\!88$$$$p^{143} T^{29} + 27322802044134664402 p^{154} T^{30} + 4036082940 p^{165} T^{31} + p^{176} T^{32}$$
53 $$1 - 511144020 T + 75439096087978483550 T^{2} -$$$$43\!\cdots\!56$$$$T^{3} +$$$$28\!\cdots\!13$$$$T^{4} +$$$$98\!\cdots\!24$$$$T^{5} +$$$$72\!\cdots\!58$$$$T^{6} +$$$$52\!\cdots\!36$$$$T^{7} +$$$$13\!\cdots\!78$$$$T^{8} +$$$$14\!\cdots\!84$$$$T^{9} +$$$$21\!\cdots\!46$$$$T^{10} +$$$$29\!\cdots\!96$$$$T^{11} +$$$$27\!\cdots\!47$$$$T^{12} +$$$$43\!\cdots\!40$$$$T^{13} +$$$$30\!\cdots\!22$$$$T^{14} +$$$$50\!\cdots\!80$$$$T^{15} +$$$$29\!\cdots\!02$$$$T^{16} +$$$$50\!\cdots\!80$$$$p^{11} T^{17} +$$$$30\!\cdots\!22$$$$p^{22} T^{18} +$$$$43\!\cdots\!40$$$$p^{33} T^{19} +$$$$27\!\cdots\!47$$$$p^{44} T^{20} +$$$$29\!\cdots\!96$$$$p^{55} T^{21} +$$$$21\!\cdots\!46$$$$p^{66} T^{22} +$$$$14\!\cdots\!84$$$$p^{77} T^{23} +$$$$13\!\cdots\!78$$$$p^{88} T^{24} +$$$$52\!\cdots\!36$$$$p^{99} T^{25} +$$$$72\!\cdots\!58$$$$p^{110} T^{26} +$$$$98\!\cdots\!24$$$$p^{121} T^{27} +$$$$28\!\cdots\!13$$$$p^{132} T^{28} -$$$$43\!\cdots\!56$$$$p^{143} T^{29} + 75439096087978483550 p^{154} T^{30} - 511144020 p^{165} T^{31} + p^{176} T^{32}$$
59 $$1 - 3728287296 T +$$$$25\!\cdots\!28$$$$T^{2} -$$$$64\!\cdots\!00$$$$T^{3} +$$$$30\!\cdots\!20$$$$T^{4} -$$$$47\!\cdots\!92$$$$T^{5} +$$$$24\!\cdots\!80$$$$T^{6} -$$$$14\!\cdots\!08$$$$T^{7} +$$$$15\!\cdots\!84$$$$T^{8} +$$$$42\!\cdots\!96$$$$T^{9} +$$$$74\!\cdots\!80$$$$T^{10} +$$$$75\!\cdots\!76$$$$T^{11} +$$$$31\!\cdots\!84$$$$T^{12} +$$$$47\!\cdots\!96$$$$T^{13} +$$$$11\!\cdots\!64$$$$T^{14} +$$$$19\!\cdots\!84$$$$T^{15} +$$$$35\!\cdots\!58$$$$T^{16} +$$$$19\!\cdots\!84$$$$p^{11} T^{17} +$$$$11\!\cdots\!64$$$$p^{22} T^{18} +$$$$47\!\cdots\!96$$$$p^{33} T^{19} +$$$$31\!\cdots\!84$$$$p^{44} T^{20} +$$$$75\!\cdots\!76$$$$p^{55} T^{21} +$$$$74\!\cdots\!80$$$$p^{66} T^{22} +$$$$42\!\cdots\!96$$$$p^{77} T^{23} +$$$$15\!\cdots\!84$$$$p^{88} T^{24} -$$$$14\!\cdots\!08$$$$p^{99} T^{25} +$$$$24\!\cdots\!80$$$$p^{110} T^{26} -$$$$47\!\cdots\!92$$$$p^{121} T^{27} +$$$$30\!\cdots\!20$$$$p^{132} T^{28} -$$$$64\!\cdots\!00$$$$p^{143} T^{29} +$$$$25\!\cdots\!28$$$$p^{154} T^{30} - 3728287296 p^{165} T^{31} + p^{176} T^{32}$$
61 $$1 - 26227205052 T +$$$$72\!\cdots\!58$$$$T^{2} -$$$$13\!\cdots\!08$$$$T^{3} +$$$$22\!\cdots\!13$$$$T^{4} -$$$$30\!\cdots\!92$$$$T^{5} +$$$$40\!\cdots\!78$$$$T^{6} -$$$$46\!\cdots\!76$$$$T^{7} +$$$$82\!\cdots\!50$$$$p T^{8} -$$$$49\!\cdots\!88$$$$T^{9} +$$$$46\!\cdots\!30$$$$T^{10} -$$$$40\!\cdots\!40$$$$T^{11} +$$$$33\!\cdots\!71$$$$T^{12} -$$$$26\!\cdots\!28$$$$T^{13} +$$$$19\!\cdots\!50$$$$T^{14} -$$$$13\!\cdots\!84$$$$T^{15} +$$$$92\!\cdots\!22$$$$T^{16} -$$$$13\!\cdots\!84$$$$p^{11} T^{17} +$$$$19\!\cdots\!50$$$$p^{22} T^{18} -$$$$26\!\cdots\!28$$$$p^{33} T^{19} +$$$$33\!\cdots\!71$$$$p^{44} T^{20} -$$$$40\!\cdots\!40$$$$p^{55} T^{21} +$$$$46\!\cdots\!30$$$$p^{66} T^{22} -$$$$49\!\cdots\!88$$$$p^{77} T^{23} +$$$$82\!\cdots\!50$$$$p^{89} T^{24} -$$$$46\!\cdots\!76$$$$p^{99} T^{25} +$$$$40\!\cdots\!78$$$$p^{110} T^{26} -$$$$30\!\cdots\!92$$$$p^{121} T^{27} +$$$$22\!\cdots\!13$$$$p^{132} T^{28} -$$$$13\!\cdots\!08$$$$p^{143} T^{29} +$$$$72\!\cdots\!58$$$$p^{154} T^{30} - 26227205052 p^{165} T^{31} + p^{176} T^{32}$$
67 $$1 + 5295680024 T +$$$$74\!\cdots\!28$$$$T^{2} +$$$$45\!\cdots\!48$$$$T^{3} +$$$$31\!\cdots\!36$$$$T^{4} +$$$$19\!\cdots\!56$$$$T^{5} +$$$$94\!\cdots\!04$$$$T^{6} +$$$$58\!\cdots\!40$$$$T^{7} +$$$$22\!\cdots\!28$$$$T^{8} +$$$$13\!\cdots\!44$$$$T^{9} +$$$$44\!\cdots\!08$$$$T^{10} +$$$$26\!\cdots\!04$$$$T^{11} +$$$$75\!\cdots\!56$$$$T^{12} +$$$$43\!\cdots\!96$$$$T^{13} +$$$$11\!\cdots\!96$$$$T^{14} +$$$$61\!\cdots\!52$$$$T^{15} +$$$$14\!\cdots\!82$$$$T^{16} +$$$$61\!\cdots\!52$$$$p^{11} T^{17} +$$$$11\!\cdots\!96$$$$p^{22} T^{18} +$$$$43\!\cdots\!96$$$$p^{33} T^{19} +$$$$75\!\cdots\!56$$$$p^{44} T^{20} +$$$$26\!\cdots\!04$$$$p^{55} T^{21} +$$$$44\!\cdots\!08$$$$p^{66} T^{22} +$$$$13\!\cdots\!44$$$$p^{77} T^{23} +$$$$22\!\cdots\!28$$$$p^{88} T^{24} +$$$$58\!\cdots\!40$$$$p^{99} T^{25} +$$$$94\!\cdots\!04$$$$p^{110} T^{26} +$$$$19\!\cdots\!56$$$$p^{121} T^{27} +$$$$31\!\cdots\!36$$$$p^{132} T^{28} +$$$$45\!\cdots\!48$$$$p^{143} T^{29} +$$$$74\!\cdots\!28$$$$p^{154} T^{30} + 5295680024 p^{165} T^{31} + p^{176} T^{32}$$
71 $$1 - 17082800928 T +$$$$16\!\cdots\!84$$$$T^{2} -$$$$26\!\cdots\!36$$$$T^{3} +$$$$13\!\cdots\!72$$$$T^{4} -$$$$22\!\cdots\!12$$$$T^{5} +$$$$73\!\cdots\!72$$$$T^{6} -$$$$13\!\cdots\!96$$$$T^{7} +$$$$32\!\cdots\!56$$$$T^{8} -$$$$59\!\cdots\!56$$$$T^{9} +$$$$11\!\cdots\!56$$$$T^{10} -$$$$21\!\cdots\!52$$$$T^{11} +$$$$37\!\cdots\!00$$$$T^{12} -$$$$63\!\cdots\!60$$$$T^{13} +$$$$10\!\cdots\!76$$$$T^{14} -$$$$16\!\cdots\!52$$$$T^{15} +$$$$26\!\cdots\!78$$$$T^{16} -$$$$16\!\cdots\!52$$$$p^{11} T^{17} +$$$$10\!\cdots\!76$$$$p^{22} T^{18} -$$$$63\!\cdots\!60$$$$p^{33} T^{19} +$$$$37\!\cdots\!00$$$$p^{44} T^{20} -$$$$21\!\cdots\!52$$$$p^{55} T^{21} +$$$$11\!\cdots\!56$$$$p^{66} T^{22} -$$$$59\!\cdots\!56$$$$p^{77} T^{23} +$$$$32\!\cdots\!56$$$$p^{88} T^{24} -$$$$13\!\cdots\!96$$$$p^{99} T^{25} +$$$$73\!\cdots\!72$$$$p^{110} T^{26} -$$$$22\!\cdots\!12$$$$p^{121} T^{27} +$$$$13\!\cdots\!72$$$$p^{132} T^{28} -$$$$26\!\cdots\!36$$$$p^{143} T^{29} +$$$$16\!\cdots\!84$$$$p^{154} T^{30} - 17082800928 p^{165} T^{31} + p^{176} T^{32}$$
73 $$1 - 21076301488 T +$$$$34\!\cdots\!80$$$$T^{2} -$$$$72\!\cdots\!48$$$$T^{3} +$$$$59\!\cdots\!50$$$$T^{4} -$$$$12\!\cdots\!92$$$$T^{5} +$$$$67\!\cdots\!20$$$$T^{6} -$$$$13\!\cdots\!44$$$$T^{7} +$$$$56\!\cdots\!29$$$$T^{8} -$$$$10\!\cdots\!16$$$$T^{9} +$$$$36\!\cdots\!52$$$$T^{10} -$$$$63\!\cdots\!12$$$$T^{11} +$$$$18\!\cdots\!18$$$$T^{12} -$$$$30\!\cdots\!76$$$$T^{13} +$$$$78\!\cdots\!36$$$$T^{14} -$$$$11\!\cdots\!44$$$$T^{15} +$$$$27\!\cdots\!52$$$$T^{16} -$$$$11\!\cdots\!44$$$$p^{11} T^{17} +$$$$78\!\cdots\!36$$$$p^{22} T^{18} -$$$$30\!\cdots\!76$$$$p^{33} T^{19} +$$$$18\!\cdots\!18$$$$p^{44} T^{20} -$$$$63\!\cdots\!12$$$$p^{55} T^{21} +$$$$36\!\cdots\!52$$$$p^{66} T^{22} -$$$$10\!\cdots\!16$$$$p^{77} T^{23} +$$$$56\!\cdots\!29$$$$p^{88} T^{24} -$$$$13\!\cdots\!44$$$$p^{99} T^{25} +$$$$67\!\cdots\!20$$$$p^{110} T^{26} -$$$$12\!\cdots\!92$$$$p^{121} T^{27} +$$$$59\!\cdots\!50$$$$p^{132} T^{28} -$$$$72\!\cdots\!48$$$$p^{143} T^{29} +$$$$34\!\cdots\!80$$$$p^{154} T^{30} - 21076301488 p^{165} T^{31} + p^{176} T^{32}$$
79 $$1 + 83677977852 T +$$$$10\!\cdots\!06$$$$T^{2} +$$$$65\!\cdots\!68$$$$T^{3} +$$$$49\!\cdots\!65$$$$T^{4} +$$$$24\!\cdots\!24$$$$T^{5} +$$$$13\!\cdots\!38$$$$T^{6} +$$$$56\!\cdots\!84$$$$T^{7} +$$$$26\!\cdots\!42$$$$T^{8} +$$$$93\!\cdots\!92$$$$T^{9} +$$$$37\!\cdots\!54$$$$T^{10} +$$$$12\!\cdots\!84$$$$T^{11} +$$$$43\!\cdots\!47$$$$T^{12} +$$$$12\!\cdots\!96$$$$T^{13} +$$$$40\!\cdots\!38$$$$T^{14} +$$$$10\!\cdots\!12$$$$T^{15} +$$$$32\!\cdots\!38$$$$T^{16} +$$$$10\!\cdots\!12$$$$p^{11} T^{17} +$$$$40\!\cdots\!38$$$$p^{22} T^{18} +$$$$12\!\cdots\!96$$$$p^{33} T^{19} +$$$$43\!\cdots\!47$$$$p^{44} T^{20} +$$$$12\!\cdots\!84$$$$p^{55} T^{21} +$$$$37\!\cdots\!54$$$$p^{66} T^{22} +$$$$93\!\cdots\!92$$$$p^{77} T^{23} +$$$$26\!\cdots\!42$$$$p^{88} T^{24} +$$$$56\!\cdots\!84$$$$p^{99} T^{25} +$$$$13\!\cdots\!38$$$$p^{110} T^{26} +$$$$24\!\cdots\!24$$$$p^{121} T^{27} +$$$$49\!\cdots\!65$$$$p^{132} T^{28} +$$$$65\!\cdots\!68$$$$p^{143} T^{29} +$$$$10\!\cdots\!06$$$$p^{154} T^{30} + 83677977852 p^{165} T^{31} + p^{176} T^{32}$$
83 $$1 + 72857072340 T +$$$$14\!\cdots\!18$$$$T^{2} +$$$$99\!\cdots\!44$$$$T^{3} +$$$$10\!\cdots\!97$$$$T^{4} +$$$$65\!\cdots\!04$$$$T^{5} +$$$$51\!\cdots\!66$$$$T^{6} +$$$$28\!\cdots\!96$$$$T^{7} +$$$$17\!\cdots\!70$$$$T^{8} +$$$$88\!\cdots\!52$$$$T^{9} +$$$$48\!\cdots\!78$$$$T^{10} +$$$$21\!\cdots\!80$$$$T^{11} +$$$$10\!\cdots\!95$$$$T^{12} +$$$$41\!\cdots\!64$$$$T^{13} +$$$$17\!\cdots\!62$$$$T^{14} +$$$$64\!\cdots\!32$$$$T^{15} +$$$$25\!\cdots\!78$$$$T^{16} +$$$$64\!\cdots\!32$$$$p^{11} T^{17} +$$$$17\!\cdots\!62$$$$p^{22} T^{18} +$$$$41\!\cdots\!64$$$$p^{33} T^{19} +$$$$10\!\cdots\!95$$$$p^{44} T^{20} +$$$$21\!\cdots\!80$$$$p^{55} T^{21} +$$$$48\!\cdots\!78$$$$p^{66} T^{22} +$$$$88\!\cdots\!52$$$$p^{77} T^{23} +$$$$17\!\cdots\!70$$$$p^{88} T^{24} +$$$$28\!\cdots\!96$$$$p^{99} T^{25} +$$$$51\!\cdots\!66$$$$p^{110} T^{26} +$$$$65\!\cdots\!04$$$$p^{121} T^{27} +$$$$10\!\cdots\!97$$$$p^{132} T^{28} +$$$$99\!\cdots\!44$$$$p^{143} T^{29} +$$$$14\!\cdots\!18$$$$p^{154} T^{30} + 72857072340 p^{165} T^{31} + p^{176} T^{32}$$
89 $$1 - 32238476676 T +$$$$21\!\cdots\!98$$$$T^{2} -$$$$83\!\cdots\!64$$$$T^{3} +$$$$24\!\cdots\!41$$$$T^{4} -$$$$10\!\cdots\!04$$$$T^{5} +$$$$18\!\cdots\!50$$$$T^{6} -$$$$88\!\cdots\!32$$$$T^{7} +$$$$11\!\cdots\!46$$$$T^{8} -$$$$55\!\cdots\!64$$$$T^{9} +$$$$51\!\cdots\!58$$$$T^{10} -$$$$27\!\cdots\!64$$$$T^{11} +$$$$20\!\cdots\!03$$$$T^{12} -$$$$10\!\cdots\!40$$$$T^{13} +$$$$67\!\cdots\!98$$$$T^{14} -$$$$36\!\cdots\!28$$$$T^{15} +$$$$20\!\cdots\!70$$$$T^{16} -$$$$36\!\cdots\!28$$$$p^{11} T^{17} +$$$$67\!\cdots\!98$$$$p^{22} T^{18} -$$$$10\!\cdots\!40$$$$p^{33} T^{19} +$$$$20\!\cdots\!03$$$$p^{44} T^{20} -$$$$27\!\cdots\!64$$$$p^{55} T^{21} +$$$$51\!\cdots\!58$$$$p^{66} T^{22} -$$$$55\!\cdots\!64$$$$p^{77} T^{23} +$$$$11\!\cdots\!46$$$$p^{88} T^{24} -$$$$88\!\cdots\!32$$$$p^{99} T^{25} +$$$$18\!\cdots\!50$$$$p^{110} T^{26} -$$$$10\!\cdots\!04$$$$p^{121} T^{27} +$$$$24\!\cdots\!41$$$$p^{132} T^{28} -$$$$83\!\cdots\!64$$$$p^{143} T^{29} +$$$$21\!\cdots\!98$$$$p^{154} T^{30} - 32238476676 p^{165} T^{31} + p^{176} T^{32}$$
97 $$1 - 6973535140 T +$$$$60\!\cdots\!74$$$$T^{2} -$$$$15\!\cdots\!88$$$$T^{3} +$$$$18\!\cdots\!45$$$$T^{4} -$$$$69\!\cdots\!08$$$$T^{5} +$$$$37\!\cdots\!34$$$$T^{6} -$$$$16\!\cdots\!60$$$$T^{7} +$$$$59\!\cdots\!90$$$$T^{8} -$$$$26\!\cdots\!88$$$$T^{9} +$$$$74\!\cdots\!38$$$$T^{10} -$$$$33\!\cdots\!52$$$$T^{11} +$$$$78\!\cdots\!35$$$$T^{12} -$$$$33\!\cdots\!28$$$$T^{13} +$$$$70\!\cdots\!98$$$$T^{14} -$$$$28\!\cdots\!32$$$$T^{15} +$$$$54\!\cdots\!38$$$$T^{16} -$$$$28\!\cdots\!32$$$$p^{11} T^{17} +$$$$70\!\cdots\!98$$$$p^{22} T^{18} -$$$$33\!\cdots\!28$$$$p^{33} T^{19} +$$$$78\!\cdots\!35$$$$p^{44} T^{20} -$$$$33\!\cdots\!52$$$$p^{55} T^{21} +$$$$74\!\cdots\!38$$$$p^{66} T^{22} -$$$$26\!\cdots\!88$$$$p^{77} T^{23} +$$$$59\!\cdots\!90$$$$p^{88} T^{24} -$$$$16\!\cdots\!60$$$$p^{99} T^{25} +$$$$37\!\cdots\!34$$$$p^{110} T^{26} -$$$$69\!\cdots\!08$$$$p^{121} T^{27} +$$$$18\!\cdots\!45$$$$p^{132} T^{28} -$$$$15\!\cdots\!88$$$$p^{143} T^{29} +$$$$60\!\cdots\!74$$$$p^{154} T^{30} - 6973535140 p^{165} T^{31} + p^{176} T^{32}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$