# Properties

 Degree $32$ Conductor $2.815\times 10^{14}$ Sign $1$ Motivic weight $17$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 270·2-s + 2.27e4·4-s + 1.15e7·7-s + 7.68e6·8-s + 7.31e8·9-s + 3.11e9·14-s + 8.63e9·16-s − 7.48e9·17-s + 1.97e11·18-s + 7.46e11·23-s + 5.19e12·25-s + 2.62e11·28-s − 3.18e11·31-s + 2.25e12·32-s − 2.02e12·34-s + 1.66e13·36-s + 7.48e12·41-s + 2.01e14·46-s − 3.76e14·47-s − 1.73e15·49-s + 1.40e15·50-s + 8.86e13·56-s − 8.61e13·62-s + 8.43e15·63-s + 4.44e14·64-s − 1.70e14·68-s + 9.02e15·71-s + ⋯
 L(s)  = 1 + 0.745·2-s + 0.173·4-s + 0.755·7-s + 0.162·8-s + 5.66·9-s + 0.563·14-s + 0.502·16-s − 0.260·17-s + 4.22·18-s + 1.98·23-s + 6.81·25-s + 0.131·28-s − 0.0671·31-s + 0.362·32-s − 0.194·34-s + 0.982·36-s + 0.146·41-s + 1.48·46-s − 2.30·47-s − 7.43·49-s + 5.08·50-s + 0.122·56-s − 0.0500·62-s + 4.28·63-s + 0.197·64-s − 0.0451·68-s + 1.65·71-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(18-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48}\right)^{s/2} \, \Gamma_{\C}(s+17/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$32$$ Conductor: $$2^{48}$$ Sign: $1$ Motivic weight: $$17$$ Character: induced by $\chi_{8} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(32,\ 2^{48} ,\ ( \ : [17/2]^{16} ),\ 1 )$$

## Particular Values

 $$L(9)$$ $$\approx$$ $$179.270$$ $$L(\frac12)$$ $$\approx$$ $$179.270$$ $$L(\frac{19}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 - 135 p T + 6271 p^{3} T^{2} - 235875 p^{6} T^{3} - 3534589 p^{10} T^{4} + 30772125 p^{15} T^{5} - 163124025 p^{21} T^{6} + 3654732165 p^{28} T^{7} - 6134904205 p^{36} T^{8} + 3654732165 p^{45} T^{9} - 163124025 p^{55} T^{10} + 30772125 p^{66} T^{11} - 3534589 p^{78} T^{12} - 235875 p^{91} T^{13} + 6271 p^{105} T^{14} - 135 p^{120} T^{15} + p^{136} T^{16}$$
good3 $$1 - 731794256 T^{2} + 31354739233887608 p^{2} T^{4} -$$$$14\!\cdots\!92$$$$p^{12} T^{6} +$$$$25\!\cdots\!00$$$$p^{10} T^{8} -$$$$50\!\cdots\!96$$$$p^{14} T^{10} +$$$$31\!\cdots\!00$$$$p^{21} T^{12} -$$$$12\!\cdots\!84$$$$p^{22} T^{14} +$$$$23\!\cdots\!46$$$$p^{30} T^{16} -$$$$12\!\cdots\!84$$$$p^{56} T^{18} +$$$$31\!\cdots\!00$$$$p^{89} T^{20} -$$$$50\!\cdots\!96$$$$p^{116} T^{22} +$$$$25\!\cdots\!00$$$$p^{146} T^{24} -$$$$14\!\cdots\!92$$$$p^{182} T^{26} + 31354739233887608 p^{206} T^{28} - 731794256 p^{238} T^{30} + p^{272} T^{32}$$
5 $$1 - 5198674409168 T^{2} +$$$$13\!\cdots\!56$$$$T^{4} -$$$$97\!\cdots\!96$$$$p^{2} T^{6} +$$$$53\!\cdots\!88$$$$p^{4} T^{8} -$$$$96\!\cdots\!88$$$$p^{8} T^{10} +$$$$14\!\cdots\!16$$$$p^{12} T^{12} -$$$$20\!\cdots\!92$$$$p^{16} T^{14} +$$$$26\!\cdots\!66$$$$p^{20} T^{16} -$$$$20\!\cdots\!92$$$$p^{50} T^{18} +$$$$14\!\cdots\!16$$$$p^{80} T^{20} -$$$$96\!\cdots\!88$$$$p^{110} T^{22} +$$$$53\!\cdots\!88$$$$p^{140} T^{24} -$$$$97\!\cdots\!96$$$$p^{172} T^{26} +$$$$13\!\cdots\!56$$$$p^{204} T^{28} - 5198674409168 p^{238} T^{30} + p^{272} T^{32}$$
7 $$( 1 - 5764800 T + 915215692443448 T^{2} -$$$$93\!\cdots\!60$$$$p T^{3} +$$$$10\!\cdots\!76$$$$p^{2} T^{4} -$$$$10\!\cdots\!60$$$$p^{3} T^{5} +$$$$11\!\cdots\!80$$$$p^{5} T^{6} -$$$$20\!\cdots\!60$$$$p^{8} T^{7} +$$$$12\!\cdots\!70$$$$p^{9} T^{8} -$$$$20\!\cdots\!60$$$$p^{25} T^{9} +$$$$11\!\cdots\!80$$$$p^{39} T^{10} -$$$$10\!\cdots\!60$$$$p^{54} T^{11} +$$$$10\!\cdots\!76$$$$p^{70} T^{12} -$$$$93\!\cdots\!60$$$$p^{86} T^{13} + 915215692443448 p^{102} T^{14} - 5764800 p^{119} T^{15} + p^{136} T^{16} )^{2}$$
11 $$1 - 3877608573076448976 T^{2} +$$$$79\!\cdots\!84$$$$T^{4} -$$$$11\!\cdots\!04$$$$T^{6} +$$$$10\!\cdots\!28$$$$p^{2} T^{8} -$$$$75\!\cdots\!92$$$$p^{4} T^{10} +$$$$46\!\cdots\!36$$$$p^{6} T^{12} -$$$$24\!\cdots\!88$$$$p^{8} T^{14} +$$$$10\!\cdots\!22$$$$p^{10} T^{16} -$$$$24\!\cdots\!88$$$$p^{42} T^{18} +$$$$46\!\cdots\!36$$$$p^{74} T^{20} -$$$$75\!\cdots\!92$$$$p^{106} T^{22} +$$$$10\!\cdots\!28$$$$p^{138} T^{24} -$$$$11\!\cdots\!04$$$$p^{170} T^{26} +$$$$79\!\cdots\!84$$$$p^{204} T^{28} - 3877608573076448976 p^{238} T^{30} + p^{272} T^{32}$$
13 $$1 - 63371249746529137488 T^{2} +$$$$20\!\cdots\!16$$$$T^{4} -$$$$28\!\cdots\!52$$$$p^{2} T^{6} +$$$$29\!\cdots\!88$$$$p^{4} T^{8} -$$$$25\!\cdots\!24$$$$p^{6} T^{10} +$$$$18\!\cdots\!84$$$$p^{8} T^{12} -$$$$11\!\cdots\!44$$$$p^{10} T^{14} +$$$$64\!\cdots\!02$$$$p^{12} T^{16} -$$$$11\!\cdots\!44$$$$p^{44} T^{18} +$$$$18\!\cdots\!84$$$$p^{76} T^{20} -$$$$25\!\cdots\!24$$$$p^{108} T^{22} +$$$$29\!\cdots\!88$$$$p^{140} T^{24} -$$$$28\!\cdots\!52$$$$p^{172} T^{26} +$$$$20\!\cdots\!16$$$$p^{204} T^{28} - 63371249746529137488 p^{238} T^{30} + p^{272} T^{32}$$
17 $$( 1 + 220268400 p T +$$$$34\!\cdots\!44$$$$T^{2} -$$$$60\!\cdots\!00$$$$p T^{3} +$$$$63\!\cdots\!60$$$$T^{4} -$$$$21\!\cdots\!00$$$$p T^{5} +$$$$81\!\cdots\!48$$$$T^{6} -$$$$30\!\cdots\!00$$$$p T^{7} +$$$$77\!\cdots\!98$$$$T^{8} -$$$$30\!\cdots\!00$$$$p^{18} T^{9} +$$$$81\!\cdots\!48$$$$p^{34} T^{10} -$$$$21\!\cdots\!00$$$$p^{52} T^{11} +$$$$63\!\cdots\!60$$$$p^{68} T^{12} -$$$$60\!\cdots\!00$$$$p^{86} T^{13} +$$$$34\!\cdots\!44$$$$p^{102} T^{14} + 220268400 p^{120} T^{15} + p^{136} T^{16} )^{2}$$
19 $$1 -$$$$50\!\cdots\!16$$$$T^{2} +$$$$12\!\cdots\!60$$$$T^{4} -$$$$21\!\cdots\!48$$$$T^{6} +$$$$27\!\cdots\!00$$$$T^{8} -$$$$27\!\cdots\!68$$$$T^{10} +$$$$22\!\cdots\!64$$$$T^{12} -$$$$15\!\cdots\!44$$$$T^{14} +$$$$94\!\cdots\!82$$$$T^{16} -$$$$15\!\cdots\!44$$$$p^{34} T^{18} +$$$$22\!\cdots\!64$$$$p^{68} T^{20} -$$$$27\!\cdots\!68$$$$p^{102} T^{22} +$$$$27\!\cdots\!00$$$$p^{136} T^{24} -$$$$21\!\cdots\!48$$$$p^{170} T^{26} +$$$$12\!\cdots\!60$$$$p^{204} T^{28} -$$$$50\!\cdots\!16$$$$p^{238} T^{30} + p^{272} T^{32}$$
23 $$( 1 - 373422672960 T +$$$$64\!\cdots\!76$$$$T^{2} -$$$$17\!\cdots\!80$$$$T^{3} +$$$$19\!\cdots\!52$$$$T^{4} -$$$$41\!\cdots\!20$$$$T^{5} +$$$$40\!\cdots\!92$$$$T^{6} -$$$$69\!\cdots\!60$$$$T^{7} +$$$$63\!\cdots\!74$$$$T^{8} -$$$$69\!\cdots\!60$$$$p^{17} T^{9} +$$$$40\!\cdots\!92$$$$p^{34} T^{10} -$$$$41\!\cdots\!20$$$$p^{51} T^{11} +$$$$19\!\cdots\!52$$$$p^{68} T^{12} -$$$$17\!\cdots\!80$$$$p^{85} T^{13} +$$$$64\!\cdots\!76$$$$p^{102} T^{14} - 373422672960 p^{119} T^{15} + p^{136} T^{16} )^{2}$$
29 $$1 -$$$$62\!\cdots\!20$$$$T^{2} +$$$$20\!\cdots\!80$$$$T^{4} -$$$$44\!\cdots\!20$$$$T^{6} +$$$$75\!\cdots\!16$$$$T^{8} -$$$$10\!\cdots\!80$$$$T^{10} +$$$$11\!\cdots\!20$$$$T^{12} -$$$$10\!\cdots\!80$$$$T^{14} +$$$$81\!\cdots\!06$$$$T^{16} -$$$$10\!\cdots\!80$$$$p^{34} T^{18} +$$$$11\!\cdots\!20$$$$p^{68} T^{20} -$$$$10\!\cdots\!80$$$$p^{102} T^{22} +$$$$75\!\cdots\!16$$$$p^{136} T^{24} -$$$$44\!\cdots\!20$$$$p^{170} T^{26} +$$$$20\!\cdots\!80$$$$p^{204} T^{28} -$$$$62\!\cdots\!20$$$$p^{238} T^{30} + p^{272} T^{32}$$
31 $$( 1 + 5144834816 p T +$$$$10\!\cdots\!04$$$$T^{2} +$$$$16\!\cdots\!28$$$$T^{3} +$$$$51\!\cdots\!12$$$$T^{4} +$$$$11\!\cdots\!72$$$$T^{5} +$$$$18\!\cdots\!80$$$$T^{6} +$$$$39\!\cdots\!28$$$$T^{7} +$$$$49\!\cdots\!98$$$$T^{8} +$$$$39\!\cdots\!28$$$$p^{17} T^{9} +$$$$18\!\cdots\!80$$$$p^{34} T^{10} +$$$$11\!\cdots\!72$$$$p^{51} T^{11} +$$$$51\!\cdots\!12$$$$p^{68} T^{12} +$$$$16\!\cdots\!28$$$$p^{85} T^{13} +$$$$10\!\cdots\!04$$$$p^{102} T^{14} + 5144834816 p^{120} T^{15} + p^{136} T^{16} )^{2}$$
37 $$1 -$$$$39\!\cdots\!44$$$$T^{2} +$$$$80\!\cdots\!76$$$$T^{4} -$$$$10\!\cdots\!72$$$$T^{6} +$$$$11\!\cdots\!64$$$$T^{8} -$$$$90\!\cdots\!04$$$$T^{10} +$$$$60\!\cdots\!24$$$$T^{12} -$$$$34\!\cdots\!60$$$$T^{14} +$$$$17\!\cdots\!90$$$$T^{16} -$$$$34\!\cdots\!60$$$$p^{34} T^{18} +$$$$60\!\cdots\!24$$$$p^{68} T^{20} -$$$$90\!\cdots\!04$$$$p^{102} T^{22} +$$$$11\!\cdots\!64$$$$p^{136} T^{24} -$$$$10\!\cdots\!72$$$$p^{170} T^{26} +$$$$80\!\cdots\!76$$$$p^{204} T^{28} -$$$$39\!\cdots\!44$$$$p^{238} T^{30} + p^{272} T^{32}$$
41 $$( 1 - 3741125768016 T +$$$$70\!\cdots\!80$$$$T^{2} -$$$$14\!\cdots\!84$$$$T^{3} +$$$$29\!\cdots\!00$$$$T^{4} +$$$$14\!\cdots\!12$$$$T^{5} +$$$$99\!\cdots\!52$$$$T^{6} +$$$$61\!\cdots\!84$$$$T^{7} +$$$$26\!\cdots\!42$$$$T^{8} +$$$$61\!\cdots\!84$$$$p^{17} T^{9} +$$$$99\!\cdots\!52$$$$p^{34} T^{10} +$$$$14\!\cdots\!12$$$$p^{51} T^{11} +$$$$29\!\cdots\!00$$$$p^{68} T^{12} -$$$$14\!\cdots\!84$$$$p^{85} T^{13} +$$$$70\!\cdots\!80$$$$p^{102} T^{14} - 3741125768016 p^{119} T^{15} + p^{136} T^{16} )^{2}$$
43 $$1 -$$$$41\!\cdots\!32$$$$T^{2} +$$$$84\!\cdots\!12$$$$T^{4} -$$$$11\!\cdots\!28$$$$T^{6} +$$$$11\!\cdots\!40$$$$T^{8} -$$$$97\!\cdots\!68$$$$T^{10} +$$$$74\!\cdots\!32$$$$T^{12} -$$$$50\!\cdots\!72$$$$T^{14} +$$$$31\!\cdots\!30$$$$T^{16} -$$$$50\!\cdots\!72$$$$p^{34} T^{18} +$$$$74\!\cdots\!32$$$$p^{68} T^{20} -$$$$97\!\cdots\!68$$$$p^{102} T^{22} +$$$$11\!\cdots\!40$$$$p^{136} T^{24} -$$$$11\!\cdots\!28$$$$p^{170} T^{26} +$$$$84\!\cdots\!12$$$$p^{204} T^{28} -$$$$41\!\cdots\!32$$$$p^{238} T^{30} + p^{272} T^{32}$$
47 $$( 1 + 188349402410880 T +$$$$19\!\cdots\!72$$$$T^{2} +$$$$31\!\cdots\!00$$$$T^{3} +$$$$17\!\cdots\!52$$$$T^{4} +$$$$23\!\cdots\!00$$$$T^{5} +$$$$87\!\cdots\!84$$$$T^{6} +$$$$99\!\cdots\!60$$$$T^{7} +$$$$28\!\cdots\!54$$$$T^{8} +$$$$99\!\cdots\!60$$$$p^{17} T^{9} +$$$$87\!\cdots\!84$$$$p^{34} T^{10} +$$$$23\!\cdots\!00$$$$p^{51} T^{11} +$$$$17\!\cdots\!52$$$$p^{68} T^{12} +$$$$31\!\cdots\!00$$$$p^{85} T^{13} +$$$$19\!\cdots\!72$$$$p^{102} T^{14} + 188349402410880 p^{119} T^{15} + p^{136} T^{16} )^{2}$$
53 $$1 -$$$$12\!\cdots\!88$$$$T^{2} +$$$$92\!\cdots\!64$$$$T^{4} -$$$$47\!\cdots\!52$$$$T^{6} +$$$$18\!\cdots\!96$$$$T^{8} -$$$$62\!\cdots\!40$$$$T^{10} +$$$$17\!\cdots\!08$$$$T^{12} -$$$$43\!\cdots\!64$$$$T^{14} +$$$$94\!\cdots\!02$$$$T^{16} -$$$$43\!\cdots\!64$$$$p^{34} T^{18} +$$$$17\!\cdots\!08$$$$p^{68} T^{20} -$$$$62\!\cdots\!40$$$$p^{102} T^{22} +$$$$18\!\cdots\!96$$$$p^{136} T^{24} -$$$$47\!\cdots\!52$$$$p^{170} T^{26} +$$$$92\!\cdots\!64$$$$p^{204} T^{28} -$$$$12\!\cdots\!88$$$$p^{238} T^{30} + p^{272} T^{32}$$
59 $$1 -$$$$20\!\cdots\!76$$$$p T^{2} +$$$$69\!\cdots\!76$$$$T^{4} -$$$$25\!\cdots\!84$$$$T^{6} +$$$$69\!\cdots\!16$$$$T^{8} -$$$$14\!\cdots\!00$$$$T^{10} +$$$$24\!\cdots\!32$$$$T^{12} -$$$$35\!\cdots\!68$$$$T^{14} +$$$$46\!\cdots\!62$$$$T^{16} -$$$$35\!\cdots\!68$$$$p^{34} T^{18} +$$$$24\!\cdots\!32$$$$p^{68} T^{20} -$$$$14\!\cdots\!00$$$$p^{102} T^{22} +$$$$69\!\cdots\!16$$$$p^{136} T^{24} -$$$$25\!\cdots\!84$$$$p^{170} T^{26} +$$$$69\!\cdots\!76$$$$p^{204} T^{28} -$$$$20\!\cdots\!76$$$$p^{239} T^{30} + p^{272} T^{32}$$
61 $$1 -$$$$16\!\cdots\!44$$$$T^{2} +$$$$12\!\cdots\!56$$$$T^{4} -$$$$64\!\cdots\!84$$$$T^{6} +$$$$24\!\cdots\!36$$$$T^{8} -$$$$82\!\cdots\!80$$$$T^{10} +$$$$24\!\cdots\!72$$$$T^{12} -$$$$64\!\cdots\!28$$$$T^{14} +$$$$15\!\cdots\!02$$$$T^{16} -$$$$64\!\cdots\!28$$$$p^{34} T^{18} +$$$$24\!\cdots\!72$$$$p^{68} T^{20} -$$$$82\!\cdots\!80$$$$p^{102} T^{22} +$$$$24\!\cdots\!36$$$$p^{136} T^{24} -$$$$64\!\cdots\!84$$$$p^{170} T^{26} +$$$$12\!\cdots\!56$$$$p^{204} T^{28} -$$$$16\!\cdots\!44$$$$p^{238} T^{30} + p^{272} T^{32}$$
67 $$1 -$$$$82\!\cdots\!12$$$$T^{2} +$$$$35\!\cdots\!44$$$$T^{4} -$$$$10\!\cdots\!88$$$$T^{6} +$$$$24\!\cdots\!96$$$$T^{8} -$$$$47\!\cdots\!60$$$$T^{10} +$$$$75\!\cdots\!48$$$$T^{12} -$$$$10\!\cdots\!76$$$$T^{14} +$$$$12\!\cdots\!82$$$$T^{16} -$$$$10\!\cdots\!76$$$$p^{34} T^{18} +$$$$75\!\cdots\!48$$$$p^{68} T^{20} -$$$$47\!\cdots\!60$$$$p^{102} T^{22} +$$$$24\!\cdots\!96$$$$p^{136} T^{24} -$$$$10\!\cdots\!88$$$$p^{170} T^{26} +$$$$35\!\cdots\!44$$$$p^{204} T^{28} -$$$$82\!\cdots\!12$$$$p^{238} T^{30} + p^{272} T^{32}$$
71 $$( 1 - 4512963142788288 T +$$$$90\!\cdots\!40$$$$T^{2} -$$$$19\!\cdots\!68$$$$T^{3} +$$$$31\!\cdots\!80$$$$T^{4} +$$$$14\!\cdots\!12$$$$T^{5} +$$$$78\!\cdots\!64$$$$T^{6} +$$$$17\!\cdots\!64$$$$T^{7} +$$$$22\!\cdots\!90$$$$T^{8} +$$$$17\!\cdots\!64$$$$p^{17} T^{9} +$$$$78\!\cdots\!64$$$$p^{34} T^{10} +$$$$14\!\cdots\!12$$$$p^{51} T^{11} +$$$$31\!\cdots\!80$$$$p^{68} T^{12} -$$$$19\!\cdots\!68$$$$p^{85} T^{13} +$$$$90\!\cdots\!40$$$$p^{102} T^{14} - 4512963142788288 p^{119} T^{15} + p^{136} T^{16} )^{2}$$
73 $$( 1 - 5666001023059280 T +$$$$17\!\cdots\!28$$$$T^{2} -$$$$10\!\cdots\!40$$$$T^{3} +$$$$12\!\cdots\!64$$$$T^{4} -$$$$66\!\cdots\!60$$$$T^{5} +$$$$61\!\cdots\!08$$$$T^{6} -$$$$24\!\cdots\!40$$$$T^{7} +$$$$27\!\cdots\!70$$$$T^{8} -$$$$24\!\cdots\!40$$$$p^{17} T^{9} +$$$$61\!\cdots\!08$$$$p^{34} T^{10} -$$$$66\!\cdots\!60$$$$p^{51} T^{11} +$$$$12\!\cdots\!64$$$$p^{68} T^{12} -$$$$10\!\cdots\!40$$$$p^{85} T^{13} +$$$$17\!\cdots\!28$$$$p^{102} T^{14} - 5666001023059280 p^{119} T^{15} + p^{136} T^{16} )^{2}$$
79 $$( 1 + 22649835696004224 T +$$$$78\!\cdots\!40$$$$T^{2} +$$$$10\!\cdots\!12$$$$T^{3} +$$$$22\!\cdots\!48$$$$T^{4} +$$$$15\!\cdots\!08$$$$T^{5} +$$$$30\!\cdots\!08$$$$T^{6} -$$$$78\!\cdots\!52$$$$T^{7} +$$$$34\!\cdots\!66$$$$T^{8} -$$$$78\!\cdots\!52$$$$p^{17} T^{9} +$$$$30\!\cdots\!08$$$$p^{34} T^{10} +$$$$15\!\cdots\!08$$$$p^{51} T^{11} +$$$$22\!\cdots\!48$$$$p^{68} T^{12} +$$$$10\!\cdots\!12$$$$p^{85} T^{13} +$$$$78\!\cdots\!40$$$$p^{102} T^{14} + 22649835696004224 p^{119} T^{15} + p^{136} T^{16} )^{2}$$
83 $$1 -$$$$26\!\cdots\!92$$$$T^{2} +$$$$34\!\cdots\!32$$$$T^{4} -$$$$29\!\cdots\!68$$$$T^{6} +$$$$21\!\cdots\!00$$$$T^{8} -$$$$13\!\cdots\!28$$$$T^{10} +$$$$71\!\cdots\!12$$$$T^{12} -$$$$34\!\cdots\!12$$$$T^{14} +$$$$15\!\cdots\!30$$$$T^{16} -$$$$34\!\cdots\!12$$$$p^{34} T^{18} +$$$$71\!\cdots\!12$$$$p^{68} T^{20} -$$$$13\!\cdots\!28$$$$p^{102} T^{22} +$$$$21\!\cdots\!00$$$$p^{136} T^{24} -$$$$29\!\cdots\!68$$$$p^{170} T^{26} +$$$$34\!\cdots\!32$$$$p^{204} T^{28} -$$$$26\!\cdots\!92$$$$p^{238} T^{30} + p^{272} T^{32}$$
89 $$( 1 + 34939587304383024 T +$$$$39\!\cdots\!00$$$$T^{2} +$$$$75\!\cdots\!32$$$$T^{3} +$$$$62\!\cdots\!48$$$$T^{4} +$$$$60\!\cdots\!48$$$$T^{5} +$$$$64\!\cdots\!08$$$$T^{6} +$$$$76\!\cdots\!28$$$$T^{7} +$$$$64\!\cdots\!66$$$$T^{8} +$$$$76\!\cdots\!28$$$$p^{17} T^{9} +$$$$64\!\cdots\!08$$$$p^{34} T^{10} +$$$$60\!\cdots\!48$$$$p^{51} T^{11} +$$$$62\!\cdots\!48$$$$p^{68} T^{12} +$$$$75\!\cdots\!32$$$$p^{85} T^{13} +$$$$39\!\cdots\!00$$$$p^{102} T^{14} + 34939587304383024 p^{119} T^{15} + p^{136} T^{16} )^{2}$$
97 $$( 1 - 47796699301090320 T +$$$$35\!\cdots\!20$$$$p T^{2} -$$$$19\!\cdots\!20$$$$T^{3} +$$$$54\!\cdots\!44$$$$T^{4} -$$$$32\!\cdots\!40$$$$T^{5} +$$$$55\!\cdots\!60$$$$T^{6} -$$$$31\!\cdots\!00$$$$T^{7} +$$$$39\!\cdots\!50$$$$T^{8} -$$$$31\!\cdots\!00$$$$p^{17} T^{9} +$$$$55\!\cdots\!60$$$$p^{34} T^{10} -$$$$32\!\cdots\!40$$$$p^{51} T^{11} +$$$$54\!\cdots\!44$$$$p^{68} T^{12} -$$$$19\!\cdots\!20$$$$p^{85} T^{13} +$$$$35\!\cdots\!20$$$$p^{103} T^{14} - 47796699301090320 p^{119} T^{15} + p^{136} T^{16} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$