# Properties

 Degree 16 Conductor $1$ Sign $1$ Motivic weight 101 Primitive no Self-dual yes Analytic rank 8

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 4.34e14·2-s − 1.20e24·3-s − 5.54e30·4-s + 3.82e34·5-s + 5.26e38·6-s − 5.78e42·7-s + 7.71e44·8-s − 2.81e48·9-s − 1.66e49·10-s + 4.62e51·11-s + 6.70e54·12-s + 2.50e56·13-s + 2.51e57·14-s − 4.62e58·15-s + 7.70e60·16-s − 3.97e62·17-s + 1.22e63·18-s − 2.13e64·19-s − 2.11e65·20-s + 6.99e66·21-s − 2.01e66·22-s − 1.36e69·23-s − 9.33e68·24-s − 1.18e71·25-s − 1.08e71·26-s + 2.63e72·27-s + 3.20e73·28-s + ⋯
 L(s)  = 1 − 0.273·2-s − 0.972·3-s − 2.18·4-s + 0.192·5-s + 0.265·6-s − 1.21·7-s + 0.191·8-s − 1.82·9-s − 0.0525·10-s + 0.118·11-s + 2.12·12-s + 1.39·13-s + 0.332·14-s − 0.187·15-s + 1.19·16-s − 2.89·17-s + 0.497·18-s − 0.566·19-s − 0.420·20-s + 1.18·21-s − 0.0324·22-s − 2.33·23-s − 0.185·24-s − 2.99·25-s − 0.380·26-s + 1.37·27-s + 2.65·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(102-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+50.5)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 $$d$$ = $$16$$ $$N$$ = $$1$$ $$\varepsilon$$ = $1$ motivic weight = $$101$$ character : $\chi_{1} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$8$$ Selberg data = $$(16,\ 1,\ (\ :[101/2]^{8}),\ 1)$$ $$L(51)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{103}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where,$$F_p(T)$$ is a polynomial of degree 16.
$p$$F_p(T)$
good2 $$1 + 13593409118595 p^{5} T +$$$$69\!\cdots\!55$$$$p^{13} T^{2} +$$$$30\!\cdots\!35$$$$p^{27} T^{3} +$$$$72\!\cdots\!13$$$$p^{45} T^{4} +$$$$15\!\cdots\!85$$$$p^{63} T^{5} +$$$$23\!\cdots\!85$$$$p^{85} T^{6} +$$$$25\!\cdots\!15$$$$p^{114} T^{7} +$$$$27\!\cdots\!39$$$$p^{146} T^{8} +$$$$25\!\cdots\!15$$$$p^{215} T^{9} +$$$$23\!\cdots\!85$$$$p^{287} T^{10} +$$$$15\!\cdots\!85$$$$p^{366} T^{11} +$$$$72\!\cdots\!13$$$$p^{449} T^{12} +$$$$30\!\cdots\!35$$$$p^{532} T^{13} +$$$$69\!\cdots\!55$$$$p^{619} T^{14} + 13593409118595 p^{712} T^{15} + p^{808} T^{16}$$
3 $$1 +$$$$13\!\cdots\!40$$$$p^{2} T +$$$$19\!\cdots\!60$$$$p^{7} T^{2} +$$$$13\!\cdots\!80$$$$p^{16} T^{3} +$$$$11\!\cdots\!28$$$$p^{27} T^{4} +$$$$12\!\cdots\!80$$$$p^{42} T^{5} +$$$$42\!\cdots\!40$$$$p^{60} T^{6} +$$$$12\!\cdots\!60$$$$p^{80} T^{7} +$$$$23\!\cdots\!18$$$$p^{103} T^{8} +$$$$12\!\cdots\!60$$$$p^{181} T^{9} +$$$$42\!\cdots\!40$$$$p^{262} T^{10} +$$$$12\!\cdots\!80$$$$p^{345} T^{11} +$$$$11\!\cdots\!28$$$$p^{431} T^{12} +$$$$13\!\cdots\!80$$$$p^{521} T^{13} +$$$$19\!\cdots\!60$$$$p^{613} T^{14} +$$$$13\!\cdots\!40$$$$p^{709} T^{15} + p^{808} T^{16}$$
5 $$1 -$$$$15\!\cdots\!88$$$$p^{2} T +$$$$15\!\cdots\!64$$$$p^{7} T^{2} +$$$$66\!\cdots\!28$$$$p^{13} T^{3} +$$$$33\!\cdots\!04$$$$p^{22} T^{4} +$$$$97\!\cdots\!48$$$$p^{32} T^{5} +$$$$38\!\cdots\!84$$$$p^{43} T^{6} +$$$$25\!\cdots\!12$$$$p^{57} T^{7} +$$$$89\!\cdots\!06$$$$p^{72} T^{8} +$$$$25\!\cdots\!12$$$$p^{158} T^{9} +$$$$38\!\cdots\!84$$$$p^{245} T^{10} +$$$$97\!\cdots\!48$$$$p^{335} T^{11} +$$$$33\!\cdots\!04$$$$p^{426} T^{12} +$$$$66\!\cdots\!28$$$$p^{518} T^{13} +$$$$15\!\cdots\!64$$$$p^{613} T^{14} -$$$$15\!\cdots\!88$$$$p^{709} T^{15} + p^{808} T^{16}$$
7 $$1 +$$$$82\!\cdots\!00$$$$p T +$$$$42\!\cdots\!00$$$$p^{4} T^{2} +$$$$72\!\cdots\!00$$$$p^{8} T^{3} +$$$$69\!\cdots\!04$$$$p^{14} T^{4} +$$$$24\!\cdots\!00$$$$p^{21} T^{5} +$$$$42\!\cdots\!00$$$$p^{29} T^{6} +$$$$22\!\cdots\!00$$$$p^{38} T^{7} +$$$$88\!\cdots\!06$$$$p^{48} T^{8} +$$$$22\!\cdots\!00$$$$p^{139} T^{9} +$$$$42\!\cdots\!00$$$$p^{231} T^{10} +$$$$24\!\cdots\!00$$$$p^{324} T^{11} +$$$$69\!\cdots\!04$$$$p^{418} T^{12} +$$$$72\!\cdots\!00$$$$p^{513} T^{13} +$$$$42\!\cdots\!00$$$$p^{610} T^{14} +$$$$82\!\cdots\!00$$$$p^{708} T^{15} + p^{808} T^{16}$$
11 $$1 -$$$$46\!\cdots\!96$$$$T +$$$$44\!\cdots\!20$$$$p^{2} T^{2} -$$$$10\!\cdots\!60$$$$p^{5} T^{3} +$$$$57\!\cdots\!20$$$$p^{9} T^{4} -$$$$17\!\cdots\!48$$$$p^{14} T^{5} +$$$$53\!\cdots\!28$$$$p^{19} T^{6} -$$$$11\!\cdots\!40$$$$p^{25} T^{7} +$$$$30\!\cdots\!70$$$$p^{32} T^{8} -$$$$11\!\cdots\!40$$$$p^{126} T^{9} +$$$$53\!\cdots\!28$$$$p^{221} T^{10} -$$$$17\!\cdots\!48$$$$p^{317} T^{11} +$$$$57\!\cdots\!20$$$$p^{413} T^{12} -$$$$10\!\cdots\!60$$$$p^{510} T^{13} +$$$$44\!\cdots\!20$$$$p^{608} T^{14} -$$$$46\!\cdots\!96$$$$p^{707} T^{15} + p^{808} T^{16}$$
13 $$1 -$$$$19\!\cdots\!60$$$$p T +$$$$80\!\cdots\!20$$$$p^{3} T^{2} -$$$$10\!\cdots\!80$$$$p^{5} T^{3} +$$$$24\!\cdots\!28$$$$p^{7} T^{4} -$$$$22\!\cdots\!40$$$$p^{10} T^{5} +$$$$16\!\cdots\!40$$$$p^{15} T^{6} -$$$$59\!\cdots\!60$$$$p^{21} T^{7} +$$$$20\!\cdots\!46$$$$p^{28} T^{8} -$$$$59\!\cdots\!60$$$$p^{122} T^{9} +$$$$16\!\cdots\!40$$$$p^{217} T^{10} -$$$$22\!\cdots\!40$$$$p^{313} T^{11} +$$$$24\!\cdots\!28$$$$p^{411} T^{12} -$$$$10\!\cdots\!80$$$$p^{510} T^{13} +$$$$80\!\cdots\!20$$$$p^{609} T^{14} -$$$$19\!\cdots\!60$$$$p^{708} T^{15} + p^{808} T^{16}$$
17 $$1 +$$$$23\!\cdots\!60$$$$p T +$$$$33\!\cdots\!60$$$$p^{3} T^{2} +$$$$28\!\cdots\!20$$$$p^{5} T^{3} +$$$$24\!\cdots\!72$$$$p^{7} T^{4} +$$$$16\!\cdots\!20$$$$p^{9} T^{5} +$$$$61\!\cdots\!60$$$$p^{12} T^{6} +$$$$64\!\cdots\!40$$$$p^{17} T^{7} +$$$$69\!\cdots\!34$$$$p^{22} T^{8} +$$$$64\!\cdots\!40$$$$p^{118} T^{9} +$$$$61\!\cdots\!60$$$$p^{214} T^{10} +$$$$16\!\cdots\!20$$$$p^{312} T^{11} +$$$$24\!\cdots\!72$$$$p^{411} T^{12} +$$$$28\!\cdots\!20$$$$p^{510} T^{13} +$$$$33\!\cdots\!60$$$$p^{609} T^{14} +$$$$23\!\cdots\!60$$$$p^{708} T^{15} + p^{808} T^{16}$$
19 $$1 +$$$$21\!\cdots\!80$$$$T +$$$$46\!\cdots\!08$$$$p T^{2} +$$$$21\!\cdots\!60$$$$p^{3} T^{3} +$$$$14\!\cdots\!92$$$$p^{5} T^{4} +$$$$28\!\cdots\!80$$$$p^{8} T^{5} +$$$$41\!\cdots\!64$$$$p^{12} T^{6} +$$$$35\!\cdots\!00$$$$p^{16} T^{7} +$$$$22\!\cdots\!30$$$$p^{21} T^{8} +$$$$35\!\cdots\!00$$$$p^{117} T^{9} +$$$$41\!\cdots\!64$$$$p^{214} T^{10} +$$$$28\!\cdots\!80$$$$p^{311} T^{11} +$$$$14\!\cdots\!92$$$$p^{409} T^{12} +$$$$21\!\cdots\!60$$$$p^{508} T^{13} +$$$$46\!\cdots\!08$$$$p^{607} T^{14} +$$$$21\!\cdots\!80$$$$p^{707} T^{15} + p^{808} T^{16}$$
23 $$1 +$$$$59\!\cdots\!60$$$$p T +$$$$32\!\cdots\!40$$$$p^{2} T^{2} +$$$$12\!\cdots\!20$$$$p^{3} T^{3} +$$$$20\!\cdots\!12$$$$p^{5} T^{4} +$$$$12\!\cdots\!60$$$$p^{8} T^{5} +$$$$74\!\cdots\!60$$$$p^{11} T^{6} +$$$$17\!\cdots\!40$$$$p^{15} T^{7} +$$$$38\!\cdots\!58$$$$p^{19} T^{8} +$$$$17\!\cdots\!40$$$$p^{116} T^{9} +$$$$74\!\cdots\!60$$$$p^{213} T^{10} +$$$$12\!\cdots\!60$$$$p^{311} T^{11} +$$$$20\!\cdots\!12$$$$p^{409} T^{12} +$$$$12\!\cdots\!20$$$$p^{508} T^{13} +$$$$32\!\cdots\!40$$$$p^{608} T^{14} +$$$$59\!\cdots\!60$$$$p^{708} T^{15} + p^{808} T^{16}$$
29 $$1 -$$$$15\!\cdots\!80$$$$T +$$$$58\!\cdots\!08$$$$p T^{2} -$$$$86\!\cdots\!40$$$$p^{2} T^{3} +$$$$19\!\cdots\!08$$$$p^{4} T^{4} -$$$$16\!\cdots\!80$$$$p^{6} T^{5} +$$$$51\!\cdots\!36$$$$p^{9} T^{6} -$$$$21\!\cdots\!00$$$$p^{12} T^{7} +$$$$40\!\cdots\!30$$$$p^{15} T^{8} -$$$$21\!\cdots\!00$$$$p^{113} T^{9} +$$$$51\!\cdots\!36$$$$p^{211} T^{10} -$$$$16\!\cdots\!80$$$$p^{309} T^{11} +$$$$19\!\cdots\!08$$$$p^{408} T^{12} -$$$$86\!\cdots\!40$$$$p^{507} T^{13} +$$$$58\!\cdots\!08$$$$p^{607} T^{14} -$$$$15\!\cdots\!80$$$$p^{707} T^{15} + p^{808} T^{16}$$
31 $$1 +$$$$65\!\cdots\!44$$$$T +$$$$10\!\cdots\!20$$$$p T^{2} +$$$$12\!\cdots\!40$$$$p^{2} T^{3} +$$$$42\!\cdots\!20$$$$p^{4} T^{4} +$$$$11\!\cdots\!72$$$$p^{6} T^{5} +$$$$32\!\cdots\!08$$$$p^{8} T^{6} +$$$$24\!\cdots\!60$$$$p^{11} T^{7} +$$$$18\!\cdots\!70$$$$p^{14} T^{8} +$$$$24\!\cdots\!60$$$$p^{112} T^{9} +$$$$32\!\cdots\!08$$$$p^{210} T^{10} +$$$$11\!\cdots\!72$$$$p^{309} T^{11} +$$$$42\!\cdots\!20$$$$p^{408} T^{12} +$$$$12\!\cdots\!40$$$$p^{507} T^{13} +$$$$10\!\cdots\!20$$$$p^{607} T^{14} +$$$$65\!\cdots\!44$$$$p^{707} T^{15} + p^{808} T^{16}$$
37 $$1 -$$$$39\!\cdots\!40$$$$T +$$$$16\!\cdots\!40$$$$T^{2} -$$$$96\!\cdots\!40$$$$p T^{3} +$$$$63\!\cdots\!04$$$$p^{2} T^{4} -$$$$26\!\cdots\!60$$$$p^{3} T^{5} +$$$$36\!\cdots\!40$$$$p^{5} T^{6} -$$$$33\!\cdots\!20$$$$p^{7} T^{7} +$$$$45\!\cdots\!58$$$$p^{9} T^{8} -$$$$33\!\cdots\!20$$$$p^{108} T^{9} +$$$$36\!\cdots\!40$$$$p^{207} T^{10} -$$$$26\!\cdots\!60$$$$p^{306} T^{11} +$$$$63\!\cdots\!04$$$$p^{406} T^{12} -$$$$96\!\cdots\!40$$$$p^{506} T^{13} +$$$$16\!\cdots\!40$$$$p^{606} T^{14} -$$$$39\!\cdots\!40$$$$p^{707} T^{15} + p^{808} T^{16}$$
41 $$1 -$$$$56\!\cdots\!36$$$$T +$$$$10\!\cdots\!20$$$$p T^{2} -$$$$10\!\cdots\!60$$$$p^{2} T^{3} +$$$$12\!\cdots\!20$$$$p^{3} T^{4} -$$$$24\!\cdots\!68$$$$p^{5} T^{5} +$$$$54\!\cdots\!28$$$$p^{7} T^{6} -$$$$92\!\cdots\!40$$$$p^{9} T^{7} +$$$$17\!\cdots\!70$$$$p^{11} T^{8} -$$$$92\!\cdots\!40$$$$p^{110} T^{9} +$$$$54\!\cdots\!28$$$$p^{209} T^{10} -$$$$24\!\cdots\!68$$$$p^{308} T^{11} +$$$$12\!\cdots\!20$$$$p^{407} T^{12} -$$$$10\!\cdots\!60$$$$p^{507} T^{13} +$$$$10\!\cdots\!20$$$$p^{607} T^{14} -$$$$56\!\cdots\!36$$$$p^{707} T^{15} + p^{808} T^{16}$$
43 $$1 +$$$$28\!\cdots\!00$$$$T +$$$$13\!\cdots\!00$$$$p T^{2} +$$$$70\!\cdots\!00$$$$p^{2} T^{3} +$$$$19\!\cdots\!28$$$$p^{3} T^{4} +$$$$78\!\cdots\!00$$$$p^{4} T^{5} +$$$$38\!\cdots\!00$$$$p^{6} T^{6} +$$$$29\!\cdots\!00$$$$p^{8} T^{7} +$$$$12\!\cdots\!94$$$$p^{10} T^{8} +$$$$29\!\cdots\!00$$$$p^{109} T^{9} +$$$$38\!\cdots\!00$$$$p^{208} T^{10} +$$$$78\!\cdots\!00$$$$p^{307} T^{11} +$$$$19\!\cdots\!28$$$$p^{407} T^{12} +$$$$70\!\cdots\!00$$$$p^{507} T^{13} +$$$$13\!\cdots\!00$$$$p^{607} T^{14} +$$$$28\!\cdots\!00$$$$p^{707} T^{15} + p^{808} T^{16}$$
47 $$1 +$$$$97\!\cdots\!40$$$$p T +$$$$19\!\cdots\!80$$$$p^{2} T^{2} +$$$$13\!\cdots\!20$$$$p^{3} T^{3} +$$$$16\!\cdots\!56$$$$p^{4} T^{4} +$$$$21\!\cdots\!40$$$$p^{6} T^{5} +$$$$41\!\cdots\!40$$$$p^{8} T^{6} +$$$$46\!\cdots\!80$$$$p^{10} T^{7} +$$$$77\!\cdots\!46$$$$p^{12} T^{8} +$$$$46\!\cdots\!80$$$$p^{111} T^{9} +$$$$41\!\cdots\!40$$$$p^{210} T^{10} +$$$$21\!\cdots\!40$$$$p^{309} T^{11} +$$$$16\!\cdots\!56$$$$p^{408} T^{12} +$$$$13\!\cdots\!20$$$$p^{508} T^{13} +$$$$19\!\cdots\!80$$$$p^{608} T^{14} +$$$$97\!\cdots\!40$$$$p^{708} T^{15} + p^{808} T^{16}$$
53 $$1 -$$$$13\!\cdots\!40$$$$T +$$$$52\!\cdots\!20$$$$T^{2} -$$$$30\!\cdots\!20$$$$T^{3} +$$$$29\!\cdots\!12$$$$p T^{4} -$$$$38\!\cdots\!20$$$$p^{2} T^{5} +$$$$23\!\cdots\!20$$$$p^{3} T^{6} -$$$$28\!\cdots\!40$$$$p^{4} T^{7} +$$$$13\!\cdots\!02$$$$p^{5} T^{8} -$$$$28\!\cdots\!40$$$$p^{105} T^{9} +$$$$23\!\cdots\!20$$$$p^{205} T^{10} -$$$$38\!\cdots\!20$$$$p^{305} T^{11} +$$$$29\!\cdots\!12$$$$p^{405} T^{12} -$$$$30\!\cdots\!20$$$$p^{505} T^{13} +$$$$52\!\cdots\!20$$$$p^{606} T^{14} -$$$$13\!\cdots\!40$$$$p^{707} T^{15} + p^{808} T^{16}$$
59 $$1 -$$$$21\!\cdots\!60$$$$T +$$$$28\!\cdots\!72$$$$T^{2} -$$$$86\!\cdots\!80$$$$T^{3} +$$$$77\!\cdots\!52$$$$p T^{4} -$$$$44\!\cdots\!60$$$$p^{2} T^{5} +$$$$24\!\cdots\!56$$$$p^{3} T^{6} -$$$$13\!\cdots\!00$$$$p^{4} T^{7} +$$$$57\!\cdots\!30$$$$p^{5} T^{8} -$$$$13\!\cdots\!00$$$$p^{105} T^{9} +$$$$24\!\cdots\!56$$$$p^{205} T^{10} -$$$$44\!\cdots\!60$$$$p^{305} T^{11} +$$$$77\!\cdots\!52$$$$p^{405} T^{12} -$$$$86\!\cdots\!80$$$$p^{505} T^{13} +$$$$28\!\cdots\!72$$$$p^{606} T^{14} -$$$$21\!\cdots\!60$$$$p^{707} T^{15} + p^{808} T^{16}$$
61 $$1 +$$$$33\!\cdots\!04$$$$T +$$$$13\!\cdots\!20$$$$T^{2} +$$$$35\!\cdots\!40$$$$T^{3} +$$$$14\!\cdots\!20$$$$p T^{4} +$$$$48\!\cdots\!92$$$$p^{2} T^{5} +$$$$14\!\cdots\!08$$$$p^{3} T^{6} +$$$$40\!\cdots\!60$$$$p^{4} T^{7} +$$$$10\!\cdots\!70$$$$p^{5} T^{8} +$$$$40\!\cdots\!60$$$$p^{105} T^{9} +$$$$14\!\cdots\!08$$$$p^{205} T^{10} +$$$$48\!\cdots\!92$$$$p^{305} T^{11} +$$$$14\!\cdots\!20$$$$p^{405} T^{12} +$$$$35\!\cdots\!40$$$$p^{505} T^{13} +$$$$13\!\cdots\!20$$$$p^{606} T^{14} +$$$$33\!\cdots\!04$$$$p^{707} T^{15} + p^{808} T^{16}$$
67 $$1 +$$$$61\!\cdots\!20$$$$T +$$$$25\!\cdots\!80$$$$T^{2} +$$$$85\!\cdots\!40$$$$T^{3} +$$$$35\!\cdots\!68$$$$p T^{4} +$$$$12\!\cdots\!60$$$$p^{2} T^{5} +$$$$41\!\cdots\!20$$$$p^{3} T^{6} +$$$$11\!\cdots\!80$$$$p^{4} T^{7} +$$$$30\!\cdots\!18$$$$p^{5} T^{8} +$$$$11\!\cdots\!80$$$$p^{105} T^{9} +$$$$41\!\cdots\!20$$$$p^{205} T^{10} +$$$$12\!\cdots\!60$$$$p^{305} T^{11} +$$$$35\!\cdots\!68$$$$p^{405} T^{12} +$$$$85\!\cdots\!40$$$$p^{505} T^{13} +$$$$25\!\cdots\!80$$$$p^{606} T^{14} +$$$$61\!\cdots\!20$$$$p^{707} T^{15} + p^{808} T^{16}$$
71 $$1 +$$$$15\!\cdots\!24$$$$T +$$$$16\!\cdots\!20$$$$T^{2} +$$$$17\!\cdots\!40$$$$p T^{3} +$$$$14\!\cdots\!20$$$$p^{2} T^{4} +$$$$10\!\cdots\!12$$$$p^{3} T^{5} +$$$$64\!\cdots\!48$$$$p^{4} T^{6} +$$$$34\!\cdots\!60$$$$p^{5} T^{7} +$$$$15\!\cdots\!70$$$$p^{6} T^{8} +$$$$34\!\cdots\!60$$$$p^{106} T^{9} +$$$$64\!\cdots\!48$$$$p^{206} T^{10} +$$$$10\!\cdots\!12$$$$p^{306} T^{11} +$$$$14\!\cdots\!20$$$$p^{406} T^{12} +$$$$17\!\cdots\!40$$$$p^{506} T^{13} +$$$$16\!\cdots\!20$$$$p^{606} T^{14} +$$$$15\!\cdots\!24$$$$p^{707} T^{15} + p^{808} T^{16}$$
73 $$1 +$$$$20\!\cdots\!80$$$$T +$$$$67\!\cdots\!60$$$$T^{2} +$$$$10\!\cdots\!80$$$$p T^{3} +$$$$28\!\cdots\!04$$$$p^{2} T^{4} +$$$$25\!\cdots\!80$$$$p^{3} T^{5} +$$$$75\!\cdots\!20$$$$p^{4} T^{6} +$$$$51\!\cdots\!60$$$$p^{5} T^{7} +$$$$21\!\cdots\!14$$$$p^{6} T^{8} +$$$$51\!\cdots\!60$$$$p^{106} T^{9} +$$$$75\!\cdots\!20$$$$p^{206} T^{10} +$$$$25\!\cdots\!80$$$$p^{306} T^{11} +$$$$28\!\cdots\!04$$$$p^{406} T^{12} +$$$$10\!\cdots\!80$$$$p^{506} T^{13} +$$$$67\!\cdots\!60$$$$p^{606} T^{14} +$$$$20\!\cdots\!80$$$$p^{707} T^{15} + p^{808} T^{16}$$
79 $$1 -$$$$14\!\cdots\!80$$$$T +$$$$34\!\cdots\!32$$$$T^{2} -$$$$47\!\cdots\!60$$$$p T^{3} +$$$$82\!\cdots\!28$$$$p^{2} T^{4} -$$$$89\!\cdots\!20$$$$p^{3} T^{5} +$$$$11\!\cdots\!64$$$$p^{4} T^{6} -$$$$10\!\cdots\!00$$$$p^{5} T^{7} +$$$$10\!\cdots\!70$$$$p^{6} T^{8} -$$$$10\!\cdots\!00$$$$p^{106} T^{9} +$$$$11\!\cdots\!64$$$$p^{206} T^{10} -$$$$89\!\cdots\!20$$$$p^{306} T^{11} +$$$$82\!\cdots\!28$$$$p^{406} T^{12} -$$$$47\!\cdots\!60$$$$p^{506} T^{13} +$$$$34\!\cdots\!32$$$$p^{606} T^{14} -$$$$14\!\cdots\!80$$$$p^{707} T^{15} + p^{808} T^{16}$$
83 $$1 -$$$$33\!\cdots\!60$$$$T +$$$$13\!\cdots\!20$$$$p^{2} T^{2} -$$$$25\!\cdots\!20$$$$p^{2} T^{3} +$$$$51\!\cdots\!88$$$$p^{3} T^{4} -$$$$81\!\cdots\!20$$$$p^{4} T^{5} +$$$$11\!\cdots\!20$$$$p^{5} T^{6} -$$$$13\!\cdots\!40$$$$p^{6} T^{7} +$$$$14\!\cdots\!38$$$$p^{7} T^{8} -$$$$13\!\cdots\!40$$$$p^{107} T^{9} +$$$$11\!\cdots\!20$$$$p^{207} T^{10} -$$$$81\!\cdots\!20$$$$p^{307} T^{11} +$$$$51\!\cdots\!88$$$$p^{407} T^{12} -$$$$25\!\cdots\!20$$$$p^{507} T^{13} +$$$$13\!\cdots\!20$$$$p^{608} T^{14} -$$$$33\!\cdots\!60$$$$p^{707} T^{15} + p^{808} T^{16}$$
89 $$1 +$$$$69\!\cdots\!40$$$$p T +$$$$55\!\cdots\!72$$$$p^{2} T^{2} +$$$$29\!\cdots\!20$$$$p^{3} T^{3} +$$$$15\!\cdots\!68$$$$p^{4} T^{4} +$$$$66\!\cdots\!40$$$$p^{5} T^{5} +$$$$27\!\cdots\!24$$$$p^{6} T^{6} +$$$$95\!\cdots\!00$$$$p^{7} T^{7} +$$$$32\!\cdots\!70$$$$p^{8} T^{8} +$$$$95\!\cdots\!00$$$$p^{108} T^{9} +$$$$27\!\cdots\!24$$$$p^{208} T^{10} +$$$$66\!\cdots\!40$$$$p^{308} T^{11} +$$$$15\!\cdots\!68$$$$p^{408} T^{12} +$$$$29\!\cdots\!20$$$$p^{508} T^{13} +$$$$55\!\cdots\!72$$$$p^{608} T^{14} +$$$$69\!\cdots\!40$$$$p^{708} T^{15} + p^{808} T^{16}$$
97 $$1 -$$$$66\!\cdots\!60$$$$p T +$$$$53\!\cdots\!80$$$$p^{2} T^{2} -$$$$23\!\cdots\!80$$$$p^{3} T^{3} +$$$$10\!\cdots\!56$$$$p^{4} T^{4} -$$$$34\!\cdots\!20$$$$p^{5} T^{5} +$$$$11\!\cdots\!60$$$$p^{6} T^{6} -$$$$28\!\cdots\!60$$$$p^{7} T^{7} +$$$$70\!\cdots\!26$$$$p^{8} T^{8} -$$$$28\!\cdots\!60$$$$p^{108} T^{9} +$$$$11\!\cdots\!60$$$$p^{208} T^{10} -$$$$34\!\cdots\!20$$$$p^{308} T^{11} +$$$$10\!\cdots\!56$$$$p^{408} T^{12} -$$$$23\!\cdots\!80$$$$p^{508} T^{13} +$$$$53\!\cdots\!80$$$$p^{608} T^{14} -$$$$66\!\cdots\!60$$$$p^{708} T^{15} + p^{808} T^{16}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}