# Properties

 Degree 36 Conductor $2^{54}$ Sign $1$ Motivic weight 20 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 398·2-s − 1.14e5·3-s + 4.41e5·4-s + 4.54e7·6-s − 1.18e9·8-s − 1.97e10·9-s + 2.78e10·11-s − 5.04e10·12-s + 2.67e11·16-s + 4.34e12·17-s + 7.86e12·18-s − 3.60e11·19-s − 1.10e13·22-s + 1.35e14·24-s + 6.60e14·25-s + 2.48e15·27-s − 9.58e14·32-s − 3.18e15·33-s − 1.72e15·34-s − 8.72e15·36-s + 1.43e14·38-s + 1.69e16·41-s − 2.74e16·43-s + 1.23e16·44-s − 3.05e16·48-s + 5.91e17·49-s − 2.62e17·50-s + ⋯
 L(s)  = 1 − 0.388·2-s − 1.93·3-s + 0.421·4-s + 0.751·6-s − 1.10·8-s − 5.66·9-s + 1.07·11-s − 0.814·12-s + 0.243·16-s + 2.15·17-s + 2.20·18-s − 0.0588·19-s − 0.417·22-s + 2.14·24-s + 6.92·25-s + 12.0·27-s − 0.851·32-s − 2.07·33-s − 0.837·34-s − 2.38·36-s + 0.0228·38-s + 1.26·41-s − 1.27·43-s + 0.452·44-s − 0.471·48-s + 7.41·49-s − 2.69·50-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{54}\right)^{s/2} \, \Gamma_{\C}(s)^{18} \, L(s)\cr =\mathstrut & \,\Lambda(21-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{54}\right)^{s/2} \, \Gamma_{\C}(s+10)^{18} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$36$$ $$N$$ = $$2^{54}$$ $$\varepsilon$$ = $1$ motivic weight = $$20$$ character : induced by $\chi_{8} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(36,\ 2^{54} ,\ ( \ : [10]^{18} ),\ 1 )$ $L(\frac{21}{2})$ $\approx$ $3.54420$ $L(\frac12)$ $\approx$ $3.54420$ $L(11)$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 2$, $$F_p$$ is a polynomial of degree 36. If $p = 2$, then $F_p$ is a polynomial of degree at most 35.
$p$$F_p$
bad2 $$1 + 199 p T - 35411 p^{3} T^{2} + 14083337 p^{6} T^{3} + 673474055 p^{10} T^{4} + 11942422939 p^{15} T^{5} + 249191446069 p^{21} T^{6} - 703784211103 p^{28} T^{7} + 7470697516139 p^{36} T^{8} + 30675285436225 p^{45} T^{9} + 7470697516139 p^{56} T^{10} - 703784211103 p^{68} T^{11} + 249191446069 p^{81} T^{12} + 11942422939 p^{95} T^{13} + 673474055 p^{110} T^{14} + 14083337 p^{126} T^{15} - 35411 p^{143} T^{16} + 199 p^{161} T^{17} + p^{180} T^{18}$$
good3 $$( 1 + 6346 p^{2} T + 4924040579 p T^{2} + 33948561926288 p^{3} T^{3} + 512002124596225436 p^{5} T^{4} +$$$$11\!\cdots\!56$$$$p^{8} T^{5} +$$$$41\!\cdots\!36$$$$p^{11} T^{6} +$$$$34\!\cdots\!28$$$$p^{19} T^{7} +$$$$86\!\cdots\!94$$$$p^{18} T^{8} +$$$$51\!\cdots\!16$$$$p^{22} T^{9} +$$$$86\!\cdots\!94$$$$p^{38} T^{10} +$$$$34\!\cdots\!28$$$$p^{59} T^{11} +$$$$41\!\cdots\!36$$$$p^{71} T^{12} +$$$$11\!\cdots\!56$$$$p^{88} T^{13} + 512002124596225436 p^{105} T^{14} + 33948561926288 p^{123} T^{15} + 4924040579 p^{141} T^{16} + 6346 p^{162} T^{17} + p^{180} T^{18} )^{2}$$
5 $$1 - 132088629252234 p T^{2} +$$$$45\!\cdots\!73$$$$p T^{4} -$$$$44\!\cdots\!52$$$$p^{3} T^{6} +$$$$67\!\cdots\!24$$$$p^{6} T^{8} -$$$$87\!\cdots\!04$$$$p^{9} T^{10} +$$$$97\!\cdots\!44$$$$p^{12} T^{12} -$$$$96\!\cdots\!24$$$$p^{15} T^{14} +$$$$86\!\cdots\!22$$$$p^{18} T^{16} -$$$$13\!\cdots\!68$$$$p^{22} T^{18} +$$$$86\!\cdots\!22$$$$p^{58} T^{20} -$$$$96\!\cdots\!24$$$$p^{95} T^{22} +$$$$97\!\cdots\!44$$$$p^{132} T^{24} -$$$$87\!\cdots\!04$$$$p^{169} T^{26} +$$$$67\!\cdots\!24$$$$p^{206} T^{28} -$$$$44\!\cdots\!52$$$$p^{243} T^{30} +$$$$45\!\cdots\!73$$$$p^{281} T^{32} - 132088629252234 p^{321} T^{34} + p^{360} T^{36}$$
7 $$1 - 1724920415944830 p^{3} T^{2} +$$$$36\!\cdots\!29$$$$p^{2} T^{4} -$$$$10\!\cdots\!40$$$$p^{3} T^{6} +$$$$24\!\cdots\!16$$$$p^{4} T^{8} -$$$$44\!\cdots\!36$$$$p^{5} T^{10} +$$$$68\!\cdots\!84$$$$p^{6} T^{12} -$$$$13\!\cdots\!72$$$$p^{8} T^{14} +$$$$47\!\cdots\!46$$$$p^{12} T^{16} -$$$$16\!\cdots\!32$$$$p^{16} T^{18} +$$$$47\!\cdots\!46$$$$p^{52} T^{20} -$$$$13\!\cdots\!72$$$$p^{88} T^{22} +$$$$68\!\cdots\!84$$$$p^{126} T^{24} -$$$$44\!\cdots\!36$$$$p^{165} T^{26} +$$$$24\!\cdots\!16$$$$p^{204} T^{28} -$$$$10\!\cdots\!40$$$$p^{243} T^{30} +$$$$36\!\cdots\!29$$$$p^{282} T^{32} - 1724920415944830 p^{323} T^{34} + p^{360} T^{36}$$
11 $$( 1 - 13931197510 T +$$$$30\!\cdots\!45$$$$T^{2} -$$$$13\!\cdots\!36$$$$p T^{3} +$$$$49\!\cdots\!80$$$$T^{4} -$$$$57\!\cdots\!60$$$$p T^{5} +$$$$47\!\cdots\!00$$$$p^{2} T^{6} +$$$$72\!\cdots\!00$$$$p^{3} T^{7} +$$$$34\!\cdots\!70$$$$p^{4} T^{8} +$$$$22\!\cdots\!40$$$$p^{5} T^{9} +$$$$34\!\cdots\!70$$$$p^{24} T^{10} +$$$$72\!\cdots\!00$$$$p^{43} T^{11} +$$$$47\!\cdots\!00$$$$p^{62} T^{12} -$$$$57\!\cdots\!60$$$$p^{81} T^{13} +$$$$49\!\cdots\!80$$$$p^{100} T^{14} -$$$$13\!\cdots\!36$$$$p^{121} T^{15} +$$$$30\!\cdots\!45$$$$p^{140} T^{16} - 13931197510 p^{160} T^{17} + p^{180} T^{18} )^{2}$$
13 $$1 -$$$$14\!\cdots\!90$$$$T^{2} +$$$$11\!\cdots\!01$$$$T^{4} -$$$$66\!\cdots\!40$$$$T^{6} +$$$$22\!\cdots\!12$$$$p T^{8} -$$$$47\!\cdots\!96$$$$p^{3} T^{10} +$$$$87\!\cdots\!12$$$$p^{5} T^{12} -$$$$13\!\cdots\!56$$$$p^{7} T^{14} +$$$$18\!\cdots\!62$$$$p^{9} T^{16} -$$$$22\!\cdots\!76$$$$p^{11} T^{18} +$$$$18\!\cdots\!62$$$$p^{49} T^{20} -$$$$13\!\cdots\!56$$$$p^{87} T^{22} +$$$$87\!\cdots\!12$$$$p^{125} T^{24} -$$$$47\!\cdots\!96$$$$p^{163} T^{26} +$$$$22\!\cdots\!12$$$$p^{201} T^{28} -$$$$66\!\cdots\!40$$$$p^{240} T^{30} +$$$$11\!\cdots\!01$$$$p^{280} T^{32} -$$$$14\!\cdots\!90$$$$p^{320} T^{34} + p^{360} T^{36}$$
17 $$( 1 - 2171962569586 T +$$$$68\!\cdots\!13$$$$p^{2} T^{2} -$$$$14\!\cdots\!36$$$$p^{2} T^{3} +$$$$41\!\cdots\!36$$$$p^{3} T^{4} -$$$$49\!\cdots\!44$$$$p^{4} T^{5} +$$$$10\!\cdots\!96$$$$p^{5} T^{6} -$$$$11\!\cdots\!56$$$$p^{6} T^{7} +$$$$18\!\cdots\!02$$$$p^{7} T^{8} -$$$$18\!\cdots\!76$$$$p^{8} T^{9} +$$$$18\!\cdots\!02$$$$p^{27} T^{10} -$$$$11\!\cdots\!56$$$$p^{46} T^{11} +$$$$10\!\cdots\!96$$$$p^{65} T^{12} -$$$$49\!\cdots\!44$$$$p^{84} T^{13} +$$$$41\!\cdots\!36$$$$p^{103} T^{14} -$$$$14\!\cdots\!36$$$$p^{122} T^{15} +$$$$68\!\cdots\!13$$$$p^{142} T^{16} - 2171962569586 p^{160} T^{17} + p^{180} T^{18} )^{2}$$
19 $$( 1 + 9491622462 p T +$$$$47\!\cdots\!17$$$$p^{2} T^{2} +$$$$31\!\cdots\!08$$$$p^{3} T^{3} +$$$$12\!\cdots\!24$$$$p^{4} T^{4} +$$$$10\!\cdots\!92$$$$p^{5} T^{5} +$$$$22\!\cdots\!20$$$$p^{6} T^{6} +$$$$18\!\cdots\!80$$$$p^{7} T^{7} +$$$$16\!\cdots\!42$$$$p^{9} T^{8} +$$$$23\!\cdots\!76$$$$p^{9} T^{9} +$$$$16\!\cdots\!42$$$$p^{29} T^{10} +$$$$18\!\cdots\!80$$$$p^{47} T^{11} +$$$$22\!\cdots\!20$$$$p^{66} T^{12} +$$$$10\!\cdots\!92$$$$p^{85} T^{13} +$$$$12\!\cdots\!24$$$$p^{104} T^{14} +$$$$31\!\cdots\!08$$$$p^{123} T^{15} +$$$$47\!\cdots\!17$$$$p^{142} T^{16} + 9491622462 p^{161} T^{17} + p^{180} T^{18} )^{2}$$
23 $$1 -$$$$18\!\cdots\!30$$$$T^{2} +$$$$71\!\cdots\!87$$$$p T^{4} -$$$$10\!\cdots\!60$$$$T^{6} +$$$$45\!\cdots\!16$$$$T^{8} -$$$$16\!\cdots\!12$$$$T^{10} +$$$$49\!\cdots\!36$$$$T^{12} -$$$$12\!\cdots\!32$$$$T^{14} +$$$$26\!\cdots\!46$$$$T^{16} -$$$$49\!\cdots\!32$$$$T^{18} +$$$$26\!\cdots\!46$$$$p^{40} T^{20} -$$$$12\!\cdots\!32$$$$p^{80} T^{22} +$$$$49\!\cdots\!36$$$$p^{120} T^{24} -$$$$16\!\cdots\!12$$$$p^{160} T^{26} +$$$$45\!\cdots\!16$$$$p^{200} T^{28} -$$$$10\!\cdots\!60$$$$p^{240} T^{30} +$$$$71\!\cdots\!87$$$$p^{281} T^{32} -$$$$18\!\cdots\!30$$$$p^{320} T^{34} + p^{360} T^{36}$$
29 $$1 -$$$$16\!\cdots\!78$$$$T^{2} +$$$$13\!\cdots\!13$$$$T^{4} -$$$$75\!\cdots\!36$$$$T^{6} +$$$$30\!\cdots\!00$$$$T^{8} -$$$$99\!\cdots\!88$$$$T^{10} +$$$$91\!\cdots\!36$$$$p T^{12} -$$$$61\!\cdots\!04$$$$T^{14} +$$$$12\!\cdots\!98$$$$T^{16} -$$$$23\!\cdots\!00$$$$T^{18} +$$$$12\!\cdots\!98$$$$p^{40} T^{20} -$$$$61\!\cdots\!04$$$$p^{80} T^{22} +$$$$91\!\cdots\!36$$$$p^{121} T^{24} -$$$$99\!\cdots\!88$$$$p^{160} T^{26} +$$$$30\!\cdots\!00$$$$p^{200} T^{28} -$$$$75\!\cdots\!36$$$$p^{240} T^{30} +$$$$13\!\cdots\!13$$$$p^{280} T^{32} -$$$$16\!\cdots\!78$$$$p^{320} T^{34} + p^{360} T^{36}$$
31 $$1 -$$$$58\!\cdots\!78$$$$T^{2} +$$$$17\!\cdots\!13$$$$T^{4} -$$$$35\!\cdots\!16$$$$T^{6} +$$$$54\!\cdots\!60$$$$T^{8} -$$$$68\!\cdots\!68$$$$T^{10} +$$$$72\!\cdots\!44$$$$T^{12} -$$$$66\!\cdots\!04$$$$T^{14} +$$$$53\!\cdots\!58$$$$T^{16} -$$$$38\!\cdots\!20$$$$T^{18} +$$$$53\!\cdots\!58$$$$p^{40} T^{20} -$$$$66\!\cdots\!04$$$$p^{80} T^{22} +$$$$72\!\cdots\!44$$$$p^{120} T^{24} -$$$$68\!\cdots\!68$$$$p^{160} T^{26} +$$$$54\!\cdots\!60$$$$p^{200} T^{28} -$$$$35\!\cdots\!16$$$$p^{240} T^{30} +$$$$17\!\cdots\!13$$$$p^{280} T^{32} -$$$$58\!\cdots\!78$$$$p^{320} T^{34} + p^{360} T^{36}$$
37 $$1 -$$$$21\!\cdots\!90$$$$T^{2} +$$$$23\!\cdots\!01$$$$T^{4} -$$$$16\!\cdots\!00$$$$T^{6} +$$$$90\!\cdots\!96$$$$T^{8} -$$$$39\!\cdots\!32$$$$T^{10} +$$$$14\!\cdots\!96$$$$T^{12} -$$$$46\!\cdots\!32$$$$T^{14} +$$$$13\!\cdots\!06$$$$T^{16} -$$$$32\!\cdots\!92$$$$T^{18} +$$$$13\!\cdots\!06$$$$p^{40} T^{20} -$$$$46\!\cdots\!32$$$$p^{80} T^{22} +$$$$14\!\cdots\!96$$$$p^{120} T^{24} -$$$$39\!\cdots\!32$$$$p^{160} T^{26} +$$$$90\!\cdots\!96$$$$p^{200} T^{28} -$$$$16\!\cdots\!00$$$$p^{240} T^{30} +$$$$23\!\cdots\!01$$$$p^{280} T^{32} -$$$$21\!\cdots\!90$$$$p^{320} T^{34} + p^{360} T^{36}$$
41 $$( 1 - 8495414878199410 T +$$$$10\!\cdots\!25$$$$T^{2} -$$$$91\!\cdots\!76$$$$T^{3} +$$$$55\!\cdots\!00$$$$T^{4} -$$$$45\!\cdots\!20$$$$T^{5} +$$$$19\!\cdots\!20$$$$T^{6} -$$$$13\!\cdots\!40$$$$T^{7} +$$$$46\!\cdots\!70$$$$T^{8} -$$$$29\!\cdots\!40$$$$T^{9} +$$$$46\!\cdots\!70$$$$p^{20} T^{10} -$$$$13\!\cdots\!40$$$$p^{40} T^{11} +$$$$19\!\cdots\!20$$$$p^{60} T^{12} -$$$$45\!\cdots\!20$$$$p^{80} T^{13} +$$$$55\!\cdots\!00$$$$p^{100} T^{14} -$$$$91\!\cdots\!76$$$$p^{120} T^{15} +$$$$10\!\cdots\!25$$$$p^{140} T^{16} - 8495414878199410 p^{160} T^{17} + p^{180} T^{18} )^{2}$$
43 $$( 1 + 13749233178376698 T +$$$$23\!\cdots\!17$$$$T^{2} +$$$$14\!\cdots\!08$$$$T^{3} +$$$$24\!\cdots\!00$$$$T^{4} -$$$$86\!\cdots\!08$$$$T^{5} +$$$$17\!\cdots\!68$$$$T^{6} -$$$$62\!\cdots\!08$$$$T^{7} +$$$$10\!\cdots\!14$$$$T^{8} -$$$$39\!\cdots\!80$$$$T^{9} +$$$$10\!\cdots\!14$$$$p^{20} T^{10} -$$$$62\!\cdots\!08$$$$p^{40} T^{11} +$$$$17\!\cdots\!68$$$$p^{60} T^{12} -$$$$86\!\cdots\!08$$$$p^{80} T^{13} +$$$$24\!\cdots\!00$$$$p^{100} T^{14} +$$$$14\!\cdots\!08$$$$p^{120} T^{15} +$$$$23\!\cdots\!17$$$$p^{140} T^{16} + 13749233178376698 p^{160} T^{17} + p^{180} T^{18} )^{2}$$
47 $$1 -$$$$28\!\cdots\!30$$$$T^{2} +$$$$41\!\cdots\!01$$$$T^{4} -$$$$41\!\cdots\!40$$$$T^{6} +$$$$31\!\cdots\!36$$$$T^{8} -$$$$18\!\cdots\!72$$$$T^{10} +$$$$90\!\cdots\!76$$$$T^{12} -$$$$37\!\cdots\!72$$$$T^{14} +$$$$13\!\cdots\!86$$$$T^{16} -$$$$38\!\cdots\!72$$$$T^{18} +$$$$13\!\cdots\!86$$$$p^{40} T^{20} -$$$$37\!\cdots\!72$$$$p^{80} T^{22} +$$$$90\!\cdots\!76$$$$p^{120} T^{24} -$$$$18\!\cdots\!72$$$$p^{160} T^{26} +$$$$31\!\cdots\!36$$$$p^{200} T^{28} -$$$$41\!\cdots\!40$$$$p^{240} T^{30} +$$$$41\!\cdots\!01$$$$p^{280} T^{32} -$$$$28\!\cdots\!30$$$$p^{320} T^{34} + p^{360} T^{36}$$
53 $$1 -$$$$12\!\cdots\!70$$$$T^{2} +$$$$94\!\cdots\!21$$$$T^{4} -$$$$54\!\cdots\!00$$$$T^{6} +$$$$25\!\cdots\!96$$$$T^{8} -$$$$10\!\cdots\!32$$$$T^{10} +$$$$36\!\cdots\!76$$$$T^{12} -$$$$12\!\cdots\!52$$$$T^{14} +$$$$38\!\cdots\!06$$$$T^{16} -$$$$12\!\cdots\!92$$$$T^{18} +$$$$38\!\cdots\!06$$$$p^{40} T^{20} -$$$$12\!\cdots\!52$$$$p^{80} T^{22} +$$$$36\!\cdots\!76$$$$p^{120} T^{24} -$$$$10\!\cdots\!32$$$$p^{160} T^{26} +$$$$25\!\cdots\!96$$$$p^{200} T^{28} -$$$$54\!\cdots\!00$$$$p^{240} T^{30} +$$$$94\!\cdots\!21$$$$p^{280} T^{32} -$$$$12\!\cdots\!70$$$$p^{320} T^{34} + p^{360} T^{36}$$
59 $$( 1 - 46556097350183110 T +$$$$19\!\cdots\!61$$$$T^{2} -$$$$56\!\cdots\!40$$$$T^{3} +$$$$17\!\cdots\!16$$$$T^{4} -$$$$31\!\cdots\!32$$$$T^{5} +$$$$97\!\cdots\!76$$$$T^{6} -$$$$11\!\cdots\!52$$$$T^{7} +$$$$36\!\cdots\!46$$$$T^{8} -$$$$30\!\cdots\!32$$$$T^{9} +$$$$36\!\cdots\!46$$$$p^{20} T^{10} -$$$$11\!\cdots\!52$$$$p^{40} T^{11} +$$$$97\!\cdots\!76$$$$p^{60} T^{12} -$$$$31\!\cdots\!32$$$$p^{80} T^{13} +$$$$17\!\cdots\!16$$$$p^{100} T^{14} -$$$$56\!\cdots\!40$$$$p^{120} T^{15} +$$$$19\!\cdots\!61$$$$p^{140} T^{16} - 46556097350183110 p^{160} T^{17} + p^{180} T^{18} )^{2}$$
61 $$1 -$$$$57\!\cdots\!98$$$$T^{2} +$$$$16\!\cdots\!93$$$$T^{4} -$$$$29\!\cdots\!36$$$$T^{6} +$$$$40\!\cdots\!80$$$$T^{8} -$$$$42\!\cdots\!88$$$$T^{10} +$$$$37\!\cdots\!64$$$$T^{12} -$$$$27\!\cdots\!84$$$$T^{14} +$$$$17\!\cdots\!98$$$$T^{16} -$$$$94\!\cdots\!60$$$$T^{18} +$$$$17\!\cdots\!98$$$$p^{40} T^{20} -$$$$27\!\cdots\!84$$$$p^{80} T^{22} +$$$$37\!\cdots\!64$$$$p^{120} T^{24} -$$$$42\!\cdots\!88$$$$p^{160} T^{26} +$$$$40\!\cdots\!80$$$$p^{200} T^{28} -$$$$29\!\cdots\!36$$$$p^{240} T^{30} +$$$$16\!\cdots\!93$$$$p^{280} T^{32} -$$$$57\!\cdots\!98$$$$p^{320} T^{34} + p^{360} T^{36}$$
67 $$( 1 + 726322821037167834 T +$$$$16\!\cdots\!97$$$$T^{2} +$$$$21\!\cdots\!76$$$$T^{3} +$$$$13\!\cdots\!28$$$$T^{4} +$$$$21\!\cdots\!56$$$$T^{5} +$$$$82\!\cdots\!52$$$$T^{6} +$$$$12\!\cdots\!76$$$$T^{7} +$$$$37\!\cdots\!66$$$$T^{8} +$$$$46\!\cdots\!24$$$$T^{9} +$$$$37\!\cdots\!66$$$$p^{20} T^{10} +$$$$12\!\cdots\!76$$$$p^{40} T^{11} +$$$$82\!\cdots\!52$$$$p^{60} T^{12} +$$$$21\!\cdots\!56$$$$p^{80} T^{13} +$$$$13\!\cdots\!28$$$$p^{100} T^{14} +$$$$21\!\cdots\!76$$$$p^{120} T^{15} +$$$$16\!\cdots\!97$$$$p^{140} T^{16} + 726322821037167834 p^{160} T^{17} + p^{180} T^{18} )^{2}$$
71 $$1 -$$$$12\!\cdots\!78$$$$T^{2} +$$$$80\!\cdots\!53$$$$T^{4} -$$$$33\!\cdots\!56$$$$T^{6} +$$$$10\!\cdots\!60$$$$T^{8} -$$$$24\!\cdots\!08$$$$T^{10} +$$$$48\!\cdots\!84$$$$T^{12} -$$$$77\!\cdots\!84$$$$T^{14} +$$$$10\!\cdots\!38$$$$T^{16} -$$$$12\!\cdots\!20$$$$T^{18} +$$$$10\!\cdots\!38$$$$p^{40} T^{20} -$$$$77\!\cdots\!84$$$$p^{80} T^{22} +$$$$48\!\cdots\!84$$$$p^{120} T^{24} -$$$$24\!\cdots\!08$$$$p^{160} T^{26} +$$$$10\!\cdots\!60$$$$p^{200} T^{28} -$$$$33\!\cdots\!56$$$$p^{240} T^{30} +$$$$80\!\cdots\!53$$$$p^{280} T^{32} -$$$$12\!\cdots\!78$$$$p^{320} T^{34} + p^{360} T^{36}$$
73 $$( 1 - 4941410656931587986 T +$$$$92\!\cdots\!77$$$$T^{2} -$$$$41\!\cdots\!64$$$$T^{3} +$$$$39\!\cdots\!88$$$$T^{4} -$$$$17\!\cdots\!04$$$$T^{5} +$$$$10\!\cdots\!72$$$$T^{6} -$$$$48\!\cdots\!24$$$$T^{7} +$$$$21\!\cdots\!46$$$$T^{8} -$$$$10\!\cdots\!96$$$$T^{9} +$$$$21\!\cdots\!46$$$$p^{20} T^{10} -$$$$48\!\cdots\!24$$$$p^{40} T^{11} +$$$$10\!\cdots\!72$$$$p^{60} T^{12} -$$$$17\!\cdots\!04$$$$p^{80} T^{13} +$$$$39\!\cdots\!88$$$$p^{100} T^{14} -$$$$41\!\cdots\!64$$$$p^{120} T^{15} +$$$$92\!\cdots\!77$$$$p^{140} T^{16} - 4941410656931587986 p^{160} T^{17} + p^{180} T^{18} )^{2}$$
79 $$1 -$$$$91\!\cdots\!18$$$$T^{2} +$$$$41\!\cdots\!73$$$$T^{4} -$$$$12\!\cdots\!16$$$$T^{6} +$$$$26\!\cdots\!20$$$$T^{8} -$$$$46\!\cdots\!68$$$$T^{10} +$$$$66\!\cdots\!84$$$$T^{12} -$$$$81\!\cdots\!64$$$$T^{14} +$$$$87\!\cdots\!58$$$$T^{16} -$$$$83\!\cdots\!40$$$$T^{18} +$$$$87\!\cdots\!58$$$$p^{40} T^{20} -$$$$81\!\cdots\!64$$$$p^{80} T^{22} +$$$$66\!\cdots\!84$$$$p^{120} T^{24} -$$$$46\!\cdots\!68$$$$p^{160} T^{26} +$$$$26\!\cdots\!20$$$$p^{200} T^{28} -$$$$12\!\cdots\!16$$$$p^{240} T^{30} +$$$$41\!\cdots\!73$$$$p^{280} T^{32} -$$$$91\!\cdots\!18$$$$p^{320} T^{34} + p^{360} T^{36}$$
83 $$( 1 + 23212764984134278874 T +$$$$98\!\cdots\!17$$$$T^{2} +$$$$92\!\cdots\!96$$$$T^{3} +$$$$33\!\cdots\!28$$$$T^{4} +$$$$13\!\cdots\!76$$$$T^{5} +$$$$12\!\cdots\!04$$$$p T^{6} +$$$$36\!\cdots\!56$$$$T^{7} +$$$$32\!\cdots\!86$$$$T^{8} +$$$$87\!\cdots\!04$$$$T^{9} +$$$$32\!\cdots\!86$$$$p^{20} T^{10} +$$$$36\!\cdots\!56$$$$p^{40} T^{11} +$$$$12\!\cdots\!04$$$$p^{61} T^{12} +$$$$13\!\cdots\!76$$$$p^{80} T^{13} +$$$$33\!\cdots\!28$$$$p^{100} T^{14} +$$$$92\!\cdots\!96$$$$p^{120} T^{15} +$$$$98\!\cdots\!17$$$$p^{140} T^{16} + 23212764984134278874 p^{160} T^{17} + p^{180} T^{18} )^{2}$$
89 $$( 1 + 14037585336125420078 T +$$$$51\!\cdots\!57$$$$T^{2} +$$$$80\!\cdots\!72$$$$T^{3} +$$$$13\!\cdots\!64$$$$T^{4} +$$$$21\!\cdots\!88$$$$T^{5} +$$$$24\!\cdots\!40$$$$T^{6} +$$$$37\!\cdots\!80$$$$T^{7} +$$$$32\!\cdots\!78$$$$T^{8} +$$$$43\!\cdots\!44$$$$T^{9} +$$$$32\!\cdots\!78$$$$p^{20} T^{10} +$$$$37\!\cdots\!80$$$$p^{40} T^{11} +$$$$24\!\cdots\!40$$$$p^{60} T^{12} +$$$$21\!\cdots\!88$$$$p^{80} T^{13} +$$$$13\!\cdots\!64$$$$p^{100} T^{14} +$$$$80\!\cdots\!72$$$$p^{120} T^{15} +$$$$51\!\cdots\!57$$$$p^{140} T^{16} + 14037585336125420078 p^{160} T^{17} + p^{180} T^{18} )^{2}$$
97 $$( 1 - 63638322586683325362 T +$$$$25\!\cdots\!97$$$$T^{2} -$$$$16\!\cdots\!32$$$$T^{3} +$$$$35\!\cdots\!00$$$$T^{4} -$$$$23\!\cdots\!88$$$$T^{5} +$$$$32\!\cdots\!88$$$$T^{6} -$$$$20\!\cdots\!68$$$$T^{7} +$$$$23\!\cdots\!14$$$$T^{8} -$$$$13\!\cdots\!00$$$$T^{9} +$$$$23\!\cdots\!14$$$$p^{20} T^{10} -$$$$20\!\cdots\!68$$$$p^{40} T^{11} +$$$$32\!\cdots\!88$$$$p^{60} T^{12} -$$$$23\!\cdots\!88$$$$p^{80} T^{13} +$$$$35\!\cdots\!00$$$$p^{100} T^{14} -$$$$16\!\cdots\!32$$$$p^{120} T^{15} +$$$$25\!\cdots\!97$$$$p^{140} T^{16} - 63638322586683325362 p^{160} T^{17} + p^{180} T^{18} )^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{36} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}