# Properties

 Degree 16 Conductor $1$ Sign $1$ Motivic weight 103 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4.38e15·2-s + 5.08e24·3-s − 6.77e30·4-s + 5.51e35·5-s + 2.23e40·6-s + 4.17e43·7-s − 3.61e46·8-s − 2.47e49·9-s + 2.42e51·10-s − 5.35e52·11-s − 3.44e55·12-s − 1.55e57·13-s + 1.83e59·14-s + 2.80e60·15-s + 5.55e61·16-s + 2.01e63·17-s − 1.08e65·18-s + 1.48e66·19-s − 3.73e66·20-s + 2.12e68·21-s − 2.35e68·22-s + 4.05e70·23-s − 1.83e71·24-s − 3.97e72·25-s − 6.80e72·26-s − 1.91e74·27-s − 2.82e74·28-s + ⋯
 L(s)  = 1 + 1.37·2-s + 1.36·3-s − 0.667·4-s + 0.555·5-s + 1.87·6-s + 1.25·7-s − 1.11·8-s − 1.77·9-s + 0.765·10-s − 0.125·11-s − 0.910·12-s − 0.664·13-s + 1.72·14-s + 0.757·15-s + 0.540·16-s + 0.863·17-s − 2.45·18-s + 2.06·19-s − 0.370·20-s + 1.70·21-s − 0.172·22-s + 3.01·23-s − 1.52·24-s − 4.02·25-s − 0.916·26-s − 3.68·27-s − 0.836·28-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut &\Gamma_{\C}(s)^{8} \, L(s)\cr =\mathstrut & \,\Lambda(104-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut &\Gamma_{\C}(s+51.5)^{8} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$16$$ $$N$$ = $$1$$ $$\varepsilon$$ = $1$ motivic weight = $$103$$ character : $\chi_{1} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(16,\ 1,\ (\ :[103/2]^{8}),\ 1)$ $L(52)$ $\approx$ $69.70511210$ $L(\frac12)$ $\approx$ $69.70511210$ $L(\frac{105}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, $$F_p$$ is a polynomial of degree 16.
$p$$F_p$
good2 $$1 - 548616194585055 p^{3} T +$$$$25\!\cdots\!75$$$$p^{10} T^{2} -$$$$51\!\cdots\!35$$$$p^{21} T^{3} +$$$$79\!\cdots\!87$$$$p^{39} T^{4} -$$$$52\!\cdots\!45$$$$p^{58} T^{5} +$$$$17\!\cdots\!75$$$$p^{78} T^{6} -$$$$20\!\cdots\!85$$$$p^{106} T^{7} +$$$$17\!\cdots\!79$$$$p^{138} T^{8} -$$$$20\!\cdots\!85$$$$p^{209} T^{9} +$$$$17\!\cdots\!75$$$$p^{284} T^{10} -$$$$52\!\cdots\!45$$$$p^{367} T^{11} +$$$$79\!\cdots\!87$$$$p^{451} T^{12} -$$$$51\!\cdots\!35$$$$p^{536} T^{13} +$$$$25\!\cdots\!75$$$$p^{628} T^{14} - 548616194585055 p^{724} T^{15} + p^{824} T^{16}$$
3 $$1 -$$$$62\!\cdots\!40$$$$p^{4} T +$$$$85\!\cdots\!00$$$$p^{10} T^{2} -$$$$49\!\cdots\!20$$$$p^{18} T^{3} +$$$$71\!\cdots\!76$$$$p^{32} T^{4} -$$$$55\!\cdots\!80$$$$p^{48} T^{5} +$$$$82\!\cdots\!00$$$$p^{66} T^{6} -$$$$82\!\cdots\!60$$$$p^{88} T^{7} +$$$$14\!\cdots\!06$$$$p^{112} T^{8} -$$$$82\!\cdots\!60$$$$p^{191} T^{9} +$$$$82\!\cdots\!00$$$$p^{272} T^{10} -$$$$55\!\cdots\!80$$$$p^{357} T^{11} +$$$$71\!\cdots\!76$$$$p^{444} T^{12} -$$$$49\!\cdots\!20$$$$p^{533} T^{13} +$$$$85\!\cdots\!00$$$$p^{628} T^{14} -$$$$62\!\cdots\!40$$$$p^{725} T^{15} + p^{824} T^{16}$$
5 $$1 -$$$$11\!\cdots\!24$$$$p T +$$$$68\!\cdots\!24$$$$p^{4} T^{2} -$$$$12\!\cdots\!84$$$$p^{9} T^{3} +$$$$12\!\cdots\!28$$$$p^{17} T^{4} -$$$$66\!\cdots\!88$$$$p^{27} T^{5} +$$$$38\!\cdots\!36$$$$p^{38} T^{6} -$$$$72\!\cdots\!24$$$$p^{50} T^{7} +$$$$29\!\cdots\!54$$$$p^{64} T^{8} -$$$$72\!\cdots\!24$$$$p^{153} T^{9} +$$$$38\!\cdots\!36$$$$p^{244} T^{10} -$$$$66\!\cdots\!88$$$$p^{336} T^{11} +$$$$12\!\cdots\!28$$$$p^{429} T^{12} -$$$$12\!\cdots\!84$$$$p^{524} T^{13} +$$$$68\!\cdots\!24$$$$p^{622} T^{14} -$$$$11\!\cdots\!24$$$$p^{722} T^{15} + p^{824} T^{16}$$
7 $$1 -$$$$85\!\cdots\!00$$$$p^{2} T +$$$$33\!\cdots\!00$$$$p^{6} T^{2} -$$$$69\!\cdots\!00$$$$p^{11} T^{3} +$$$$47\!\cdots\!04$$$$p^{18} T^{4} -$$$$39\!\cdots\!00$$$$p^{27} T^{5} +$$$$39\!\cdots\!00$$$$p^{36} T^{6} -$$$$33\!\cdots\!00$$$$p^{45} T^{7} +$$$$55\!\cdots\!06$$$$p^{56} T^{8} -$$$$33\!\cdots\!00$$$$p^{148} T^{9} +$$$$39\!\cdots\!00$$$$p^{242} T^{10} -$$$$39\!\cdots\!00$$$$p^{336} T^{11} +$$$$47\!\cdots\!04$$$$p^{430} T^{12} -$$$$69\!\cdots\!00$$$$p^{526} T^{13} +$$$$33\!\cdots\!00$$$$p^{624} T^{14} -$$$$85\!\cdots\!00$$$$p^{723} T^{15} + p^{824} T^{16}$$
11 $$1 +$$$$48\!\cdots\!04$$$$p T +$$$$79\!\cdots\!20$$$$p^{2} T^{2} +$$$$57\!\cdots\!40$$$$p^{5} T^{3} +$$$$19\!\cdots\!20$$$$p^{9} T^{4} +$$$$14\!\cdots\!52$$$$p^{14} T^{5} +$$$$20\!\cdots\!28$$$$p^{20} T^{6} +$$$$14\!\cdots\!60$$$$p^{26} T^{7} +$$$$12\!\cdots\!70$$$$p^{33} T^{8} +$$$$14\!\cdots\!60$$$$p^{129} T^{9} +$$$$20\!\cdots\!28$$$$p^{226} T^{10} +$$$$14\!\cdots\!52$$$$p^{323} T^{11} +$$$$19\!\cdots\!20$$$$p^{421} T^{12} +$$$$57\!\cdots\!40$$$$p^{520} T^{13} +$$$$79\!\cdots\!20$$$$p^{620} T^{14} +$$$$48\!\cdots\!04$$$$p^{722} T^{15} + p^{824} T^{16}$$
13 $$1 +$$$$11\!\cdots\!40$$$$p T +$$$$14\!\cdots\!00$$$$p^{3} T^{2} +$$$$59\!\cdots\!60$$$$p^{6} T^{3} +$$$$43\!\cdots\!32$$$$p^{9} T^{4} +$$$$97\!\cdots\!40$$$$p^{12} T^{5} +$$$$61\!\cdots\!00$$$$p^{16} T^{6} +$$$$49\!\cdots\!60$$$$p^{21} T^{7} +$$$$16\!\cdots\!66$$$$p^{28} T^{8} +$$$$49\!\cdots\!60$$$$p^{124} T^{9} +$$$$61\!\cdots\!00$$$$p^{222} T^{10} +$$$$97\!\cdots\!40$$$$p^{321} T^{11} +$$$$43\!\cdots\!32$$$$p^{421} T^{12} +$$$$59\!\cdots\!60$$$$p^{521} T^{13} +$$$$14\!\cdots\!00$$$$p^{621} T^{14} +$$$$11\!\cdots\!40$$$$p^{722} T^{15} + p^{824} T^{16}$$
17 $$1 -$$$$11\!\cdots\!60$$$$p T +$$$$55\!\cdots\!00$$$$p^{2} T^{2} -$$$$20\!\cdots\!60$$$$p^{4} T^{3} +$$$$30\!\cdots\!12$$$$p^{7} T^{4} -$$$$31\!\cdots\!60$$$$p^{10} T^{5} +$$$$78\!\cdots\!00$$$$p^{13} T^{6} -$$$$24\!\cdots\!20$$$$p^{16} T^{7} +$$$$10\!\cdots\!66$$$$p^{20} T^{8} -$$$$24\!\cdots\!20$$$$p^{119} T^{9} +$$$$78\!\cdots\!00$$$$p^{219} T^{10} -$$$$31\!\cdots\!60$$$$p^{319} T^{11} +$$$$30\!\cdots\!12$$$$p^{419} T^{12} -$$$$20\!\cdots\!60$$$$p^{519} T^{13} +$$$$55\!\cdots\!00$$$$p^{620} T^{14} -$$$$11\!\cdots\!60$$$$p^{722} T^{15} + p^{824} T^{16}$$
19 $$1 -$$$$78\!\cdots\!00$$$$p T +$$$$71\!\cdots\!52$$$$p^{2} T^{2} -$$$$16\!\cdots\!00$$$$p^{4} T^{3} +$$$$21\!\cdots\!12$$$$p^{7} T^{4} -$$$$98\!\cdots\!00$$$$p^{10} T^{5} +$$$$15\!\cdots\!44$$$$p^{14} T^{6} +$$$$26\!\cdots\!00$$$$p^{19} T^{7} -$$$$66\!\cdots\!30$$$$p^{24} T^{8} +$$$$26\!\cdots\!00$$$$p^{122} T^{9} +$$$$15\!\cdots\!44$$$$p^{220} T^{10} -$$$$98\!\cdots\!00$$$$p^{319} T^{11} +$$$$21\!\cdots\!12$$$$p^{419} T^{12} -$$$$16\!\cdots\!00$$$$p^{519} T^{13} +$$$$71\!\cdots\!52$$$$p^{620} T^{14} -$$$$78\!\cdots\!00$$$$p^{722} T^{15} + p^{824} T^{16}$$
23 $$1 -$$$$40\!\cdots\!20$$$$T +$$$$65\!\cdots\!00$$$$p T^{2} -$$$$26\!\cdots\!20$$$$p^{3} T^{3} +$$$$98\!\cdots\!92$$$$p^{5} T^{4} -$$$$24\!\cdots\!20$$$$p^{7} T^{5} +$$$$28\!\cdots\!00$$$$p^{10} T^{6} -$$$$94\!\cdots\!20$$$$p^{14} T^{7} +$$$$50\!\cdots\!54$$$$p^{18} T^{8} -$$$$94\!\cdots\!20$$$$p^{117} T^{9} +$$$$28\!\cdots\!00$$$$p^{216} T^{10} -$$$$24\!\cdots\!20$$$$p^{316} T^{11} +$$$$98\!\cdots\!92$$$$p^{417} T^{12} -$$$$26\!\cdots\!20$$$$p^{518} T^{13} +$$$$65\!\cdots\!00$$$$p^{619} T^{14} -$$$$40\!\cdots\!20$$$$p^{721} T^{15} + p^{824} T^{16}$$
29 $$1 +$$$$11\!\cdots\!00$$$$T +$$$$82\!\cdots\!28$$$$p T^{2} +$$$$37\!\cdots\!00$$$$p^{2} T^{3} +$$$$11\!\cdots\!92$$$$p^{3} T^{4} +$$$$18\!\cdots\!00$$$$p^{5} T^{5} +$$$$11\!\cdots\!96$$$$p^{7} T^{6} +$$$$60\!\cdots\!00$$$$p^{10} T^{7} +$$$$95\!\cdots\!30$$$$p^{13} T^{8} +$$$$60\!\cdots\!00$$$$p^{113} T^{9} +$$$$11\!\cdots\!96$$$$p^{213} T^{10} +$$$$18\!\cdots\!00$$$$p^{314} T^{11} +$$$$11\!\cdots\!92$$$$p^{415} T^{12} +$$$$37\!\cdots\!00$$$$p^{517} T^{13} +$$$$82\!\cdots\!28$$$$p^{619} T^{14} +$$$$11\!\cdots\!00$$$$p^{721} T^{15} + p^{824} T^{16}$$
31 $$1 -$$$$33\!\cdots\!56$$$$p T +$$$$26\!\cdots\!20$$$$p^{2} T^{2} -$$$$70\!\cdots\!60$$$$p^{3} T^{3} +$$$$10\!\cdots\!20$$$$p^{5} T^{4} -$$$$74\!\cdots\!88$$$$p^{7} T^{5} +$$$$27\!\cdots\!68$$$$p^{10} T^{6} -$$$$51\!\cdots\!40$$$$p^{13} T^{7} +$$$$14\!\cdots\!70$$$$p^{16} T^{8} -$$$$51\!\cdots\!40$$$$p^{116} T^{9} +$$$$27\!\cdots\!68$$$$p^{216} T^{10} -$$$$74\!\cdots\!88$$$$p^{316} T^{11} +$$$$10\!\cdots\!20$$$$p^{417} T^{12} -$$$$70\!\cdots\!60$$$$p^{518} T^{13} +$$$$26\!\cdots\!20$$$$p^{620} T^{14} -$$$$33\!\cdots\!56$$$$p^{722} T^{15} + p^{824} T^{16}$$
37 $$1 +$$$$92\!\cdots\!40$$$$T +$$$$12\!\cdots\!00$$$$T^{2} +$$$$57\!\cdots\!60$$$$p T^{3} +$$$$40\!\cdots\!44$$$$p^{2} T^{4} +$$$$39\!\cdots\!60$$$$p^{3} T^{5} +$$$$13\!\cdots\!00$$$$p^{5} T^{6} +$$$$11\!\cdots\!80$$$$p^{7} T^{7} +$$$$29\!\cdots\!18$$$$p^{9} T^{8} +$$$$11\!\cdots\!80$$$$p^{110} T^{9} +$$$$13\!\cdots\!00$$$$p^{211} T^{10} +$$$$39\!\cdots\!60$$$$p^{312} T^{11} +$$$$40\!\cdots\!44$$$$p^{414} T^{12} +$$$$57\!\cdots\!60$$$$p^{516} T^{13} +$$$$12\!\cdots\!00$$$$p^{618} T^{14} +$$$$92\!\cdots\!40$$$$p^{721} T^{15} + p^{824} T^{16}$$
41 $$1 -$$$$22\!\cdots\!76$$$$T +$$$$64\!\cdots\!20$$$$T^{2} -$$$$22\!\cdots\!60$$$$p T^{3} +$$$$10\!\cdots\!20$$$$p^{2} T^{4} -$$$$33\!\cdots\!08$$$$p^{3} T^{5} +$$$$32\!\cdots\!88$$$$p^{5} T^{6} -$$$$21\!\cdots\!40$$$$p^{7} T^{7} +$$$$17\!\cdots\!70$$$$p^{9} T^{8} -$$$$21\!\cdots\!40$$$$p^{110} T^{9} +$$$$32\!\cdots\!88$$$$p^{211} T^{10} -$$$$33\!\cdots\!08$$$$p^{312} T^{11} +$$$$10\!\cdots\!20$$$$p^{414} T^{12} -$$$$22\!\cdots\!60$$$$p^{516} T^{13} +$$$$64\!\cdots\!20$$$$p^{618} T^{14} -$$$$22\!\cdots\!76$$$$p^{721} T^{15} + p^{824} T^{16}$$
43 $$1 -$$$$96\!\cdots\!00$$$$T +$$$$79\!\cdots\!00$$$$p T^{2} -$$$$16\!\cdots\!00$$$$p^{2} T^{3} +$$$$67\!\cdots\!28$$$$p^{3} T^{4} -$$$$30\!\cdots\!00$$$$p^{5} T^{5} +$$$$53\!\cdots\!00$$$$p^{7} T^{6} -$$$$28\!\cdots\!00$$$$p^{9} T^{7} +$$$$39\!\cdots\!58$$$$p^{11} T^{8} -$$$$28\!\cdots\!00$$$$p^{112} T^{9} +$$$$53\!\cdots\!00$$$$p^{213} T^{10} -$$$$30\!\cdots\!00$$$$p^{314} T^{11} +$$$$67\!\cdots\!28$$$$p^{415} T^{12} -$$$$16\!\cdots\!00$$$$p^{517} T^{13} +$$$$79\!\cdots\!00$$$$p^{619} T^{14} -$$$$96\!\cdots\!00$$$$p^{721} T^{15} + p^{824} T^{16}$$
47 $$1 +$$$$29\!\cdots\!20$$$$T +$$$$22\!\cdots\!00$$$$p T^{2} +$$$$10\!\cdots\!40$$$$p^{2} T^{3} +$$$$49\!\cdots\!92$$$$p^{3} T^{4} +$$$$37\!\cdots\!20$$$$p^{5} T^{5} +$$$$29\!\cdots\!00$$$$p^{7} T^{6} +$$$$18\!\cdots\!60$$$$p^{9} T^{7} +$$$$11\!\cdots\!82$$$$p^{11} T^{8} +$$$$18\!\cdots\!60$$$$p^{112} T^{9} +$$$$29\!\cdots\!00$$$$p^{213} T^{10} +$$$$37\!\cdots\!20$$$$p^{314} T^{11} +$$$$49\!\cdots\!92$$$$p^{415} T^{12} +$$$$10\!\cdots\!40$$$$p^{517} T^{13} +$$$$22\!\cdots\!00$$$$p^{619} T^{14} +$$$$29\!\cdots\!20$$$$p^{721} T^{15} + p^{824} T^{16}$$
53 $$1 -$$$$16\!\cdots\!40$$$$T +$$$$29\!\cdots\!00$$$$T^{2} -$$$$32\!\cdots\!80$$$$T^{3} +$$$$37\!\cdots\!16$$$$T^{4} -$$$$60\!\cdots\!60$$$$p T^{5} +$$$$99\!\cdots\!00$$$$p^{2} T^{6} -$$$$12\!\cdots\!80$$$$p^{3} T^{7} +$$$$17\!\cdots\!66$$$$p^{4} T^{8} -$$$$12\!\cdots\!80$$$$p^{106} T^{9} +$$$$99\!\cdots\!00$$$$p^{208} T^{10} -$$$$60\!\cdots\!60$$$$p^{310} T^{11} +$$$$37\!\cdots\!16$$$$p^{412} T^{12} -$$$$32\!\cdots\!80$$$$p^{515} T^{13} +$$$$29\!\cdots\!00$$$$p^{618} T^{14} -$$$$16\!\cdots\!40$$$$p^{721} T^{15} + p^{824} T^{16}$$
59 $$1 +$$$$37\!\cdots\!00$$$$T +$$$$13\!\cdots\!32$$$$T^{2} +$$$$51\!\cdots\!00$$$$p T^{3} +$$$$14\!\cdots\!72$$$$p T^{4} +$$$$36\!\cdots\!00$$$$p^{2} T^{5} +$$$$16\!\cdots\!96$$$$p^{3} T^{6} +$$$$33\!\cdots\!00$$$$p^{4} T^{7} +$$$$14\!\cdots\!30$$$$p^{5} T^{8} +$$$$33\!\cdots\!00$$$$p^{107} T^{9} +$$$$16\!\cdots\!96$$$$p^{209} T^{10} +$$$$36\!\cdots\!00$$$$p^{311} T^{11} +$$$$14\!\cdots\!72$$$$p^{413} T^{12} +$$$$51\!\cdots\!00$$$$p^{516} T^{13} +$$$$13\!\cdots\!32$$$$p^{618} T^{14} +$$$$37\!\cdots\!00$$$$p^{721} T^{15} + p^{824} T^{16}$$
61 $$1 -$$$$56\!\cdots\!56$$$$T +$$$$45\!\cdots\!20$$$$T^{2} -$$$$22\!\cdots\!60$$$$T^{3} +$$$$16\!\cdots\!20$$$$p T^{4} -$$$$11\!\cdots\!08$$$$p^{2} T^{5} +$$$$59\!\cdots\!88$$$$p^{3} T^{6} -$$$$36\!\cdots\!40$$$$p^{4} T^{7} +$$$$14\!\cdots\!70$$$$p^{5} T^{8} -$$$$36\!\cdots\!40$$$$p^{107} T^{9} +$$$$59\!\cdots\!88$$$$p^{209} T^{10} -$$$$11\!\cdots\!08$$$$p^{311} T^{11} +$$$$16\!\cdots\!20$$$$p^{413} T^{12} -$$$$22\!\cdots\!60$$$$p^{515} T^{13} +$$$$45\!\cdots\!20$$$$p^{618} T^{14} -$$$$56\!\cdots\!56$$$$p^{721} T^{15} + p^{824} T^{16}$$
67 $$1 -$$$$36\!\cdots\!20$$$$T +$$$$11\!\cdots\!00$$$$T^{2} -$$$$34\!\cdots\!80$$$$p T^{3} +$$$$97\!\cdots\!84$$$$p^{2} T^{4} -$$$$21\!\cdots\!80$$$$p^{3} T^{5} +$$$$45\!\cdots\!00$$$$p^{4} T^{6} -$$$$79\!\cdots\!60$$$$p^{5} T^{7} +$$$$14\!\cdots\!14$$$$p^{6} T^{8} -$$$$79\!\cdots\!60$$$$p^{108} T^{9} +$$$$45\!\cdots\!00$$$$p^{210} T^{10} -$$$$21\!\cdots\!80$$$$p^{312} T^{11} +$$$$97\!\cdots\!84$$$$p^{414} T^{12} -$$$$34\!\cdots\!80$$$$p^{516} T^{13} +$$$$11\!\cdots\!00$$$$p^{618} T^{14} -$$$$36\!\cdots\!20$$$$p^{721} T^{15} + p^{824} T^{16}$$
71 $$1 -$$$$13\!\cdots\!96$$$$T +$$$$10\!\cdots\!20$$$$T^{2} -$$$$85\!\cdots\!60$$$$p T^{3} +$$$$54\!\cdots\!20$$$$p^{2} T^{4} -$$$$28\!\cdots\!88$$$$p^{3} T^{5} +$$$$12\!\cdots\!08$$$$p^{4} T^{6} -$$$$48\!\cdots\!40$$$$p^{5} T^{7} +$$$$16\!\cdots\!70$$$$p^{6} T^{8} -$$$$48\!\cdots\!40$$$$p^{108} T^{9} +$$$$12\!\cdots\!08$$$$p^{210} T^{10} -$$$$28\!\cdots\!88$$$$p^{312} T^{11} +$$$$54\!\cdots\!20$$$$p^{414} T^{12} -$$$$85\!\cdots\!60$$$$p^{516} T^{13} +$$$$10\!\cdots\!20$$$$p^{618} T^{14} -$$$$13\!\cdots\!96$$$$p^{721} T^{15} + p^{824} T^{16}$$
73 $$1 -$$$$44\!\cdots\!20$$$$T +$$$$10\!\cdots\!00$$$$T^{2} -$$$$27\!\cdots\!80$$$$p T^{3} +$$$$54\!\cdots\!64$$$$p^{2} T^{4} -$$$$95\!\cdots\!20$$$$p^{3} T^{5} +$$$$15\!\cdots\!00$$$$p^{4} T^{6} -$$$$22\!\cdots\!60$$$$p^{5} T^{7} +$$$$29\!\cdots\!34$$$$p^{6} T^{8} -$$$$22\!\cdots\!60$$$$p^{108} T^{9} +$$$$15\!\cdots\!00$$$$p^{210} T^{10} -$$$$95\!\cdots\!20$$$$p^{312} T^{11} +$$$$54\!\cdots\!64$$$$p^{414} T^{12} -$$$$27\!\cdots\!80$$$$p^{516} T^{13} +$$$$10\!\cdots\!00$$$$p^{618} T^{14} -$$$$44\!\cdots\!20$$$$p^{721} T^{15} + p^{824} T^{16}$$
79 $$1 -$$$$24\!\cdots\!00$$$$T +$$$$15\!\cdots\!28$$$$p T^{2} -$$$$54\!\cdots\!00$$$$p^{2} T^{3} +$$$$15\!\cdots\!92$$$$p^{3} T^{4} -$$$$54\!\cdots\!00$$$$p^{4} T^{5} +$$$$10\!\cdots\!36$$$$p^{5} T^{6} -$$$$34\!\cdots\!00$$$$p^{6} T^{7} +$$$$52\!\cdots\!30$$$$p^{7} T^{8} -$$$$34\!\cdots\!00$$$$p^{109} T^{9} +$$$$10\!\cdots\!36$$$$p^{211} T^{10} -$$$$54\!\cdots\!00$$$$p^{313} T^{11} +$$$$15\!\cdots\!92$$$$p^{415} T^{12} -$$$$54\!\cdots\!00$$$$p^{517} T^{13} +$$$$15\!\cdots\!28$$$$p^{619} T^{14} -$$$$24\!\cdots\!00$$$$p^{721} T^{15} + p^{824} T^{16}$$
83 $$1 +$$$$25\!\cdots\!40$$$$T +$$$$63\!\cdots\!00$$$$p T^{2} +$$$$10\!\cdots\!20$$$$p^{2} T^{3} +$$$$16\!\cdots\!48$$$$p^{3} T^{4} +$$$$19\!\cdots\!80$$$$p^{4} T^{5} +$$$$22\!\cdots\!00$$$$p^{5} T^{6} +$$$$21\!\cdots\!40$$$$p^{6} T^{7} +$$$$18\!\cdots\!58$$$$p^{7} T^{8} +$$$$21\!\cdots\!40$$$$p^{109} T^{9} +$$$$22\!\cdots\!00$$$$p^{211} T^{10} +$$$$19\!\cdots\!80$$$$p^{313} T^{11} +$$$$16\!\cdots\!48$$$$p^{415} T^{12} +$$$$10\!\cdots\!20$$$$p^{517} T^{13} +$$$$63\!\cdots\!00$$$$p^{619} T^{14} +$$$$25\!\cdots\!40$$$$p^{721} T^{15} + p^{824} T^{16}$$
89 $$1 +$$$$24\!\cdots\!00$$$$T +$$$$28\!\cdots\!68$$$$p T^{2} -$$$$25\!\cdots\!00$$$$p^{2} T^{3} +$$$$52\!\cdots\!32$$$$p^{3} T^{4} -$$$$61\!\cdots\!00$$$$p^{4} T^{5} +$$$$65\!\cdots\!96$$$$p^{5} T^{6} -$$$$91\!\cdots\!00$$$$p^{6} T^{7} +$$$$58\!\cdots\!30$$$$p^{7} T^{8} -$$$$91\!\cdots\!00$$$$p^{109} T^{9} +$$$$65\!\cdots\!96$$$$p^{211} T^{10} -$$$$61\!\cdots\!00$$$$p^{313} T^{11} +$$$$52\!\cdots\!32$$$$p^{415} T^{12} -$$$$25\!\cdots\!00$$$$p^{517} T^{13} +$$$$28\!\cdots\!68$$$$p^{619} T^{14} +$$$$24\!\cdots\!00$$$$p^{721} T^{15} + p^{824} T^{16}$$
97 $$1 -$$$$51\!\cdots\!40$$$$p T +$$$$39\!\cdots\!00$$$$p^{2} T^{2} -$$$$14\!\cdots\!80$$$$p^{3} T^{3} +$$$$61\!\cdots\!36$$$$p^{4} T^{4} -$$$$17\!\cdots\!80$$$$p^{5} T^{5} +$$$$54\!\cdots\!00$$$$p^{6} T^{6} -$$$$12\!\cdots\!60$$$$p^{7} T^{7} +$$$$30\!\cdots\!86$$$$p^{8} T^{8} -$$$$12\!\cdots\!60$$$$p^{110} T^{9} +$$$$54\!\cdots\!00$$$$p^{212} T^{10} -$$$$17\!\cdots\!80$$$$p^{314} T^{11} +$$$$61\!\cdots\!36$$$$p^{416} T^{12} -$$$$14\!\cdots\!80$$$$p^{518} T^{13} +$$$$39\!\cdots\!00$$$$p^{620} T^{14} -$$$$51\!\cdots\!40$$$$p^{722} T^{15} + p^{824} T^{16}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}