# Properties

 Degree 62 Conductor $3^{31} \cdot 17^{31} \cdot 79^{31}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4·2-s + 31·3-s − 6·4-s + 11·5-s + 124·6-s + 4·7-s − 44·8-s + 496·9-s + 44·10-s + 26·11-s − 186·12-s + 7·13-s + 16·14-s + 341·15-s − 5·16-s + 31·17-s + 1.98e3·18-s + 32·19-s − 66·20-s + 124·21-s + 104·22-s + 29·23-s − 1.36e3·24-s − 25-s + 28·26-s + 5.45e3·27-s − 24·28-s + ⋯
 L(s)  = 1 + 2.82·2-s + 17.8·3-s − 3·4-s + 4.91·5-s + 50.6·6-s + 1.51·7-s − 15.5·8-s + 165.·9-s + 13.9·10-s + 7.83·11-s − 53.6·12-s + 1.94·13-s + 4.27·14-s + 88.0·15-s − 5/4·16-s + 7.51·17-s + 467.·18-s + 7.34·19-s − 14.7·20-s + 27.0·21-s + 22.1·22-s + 6.04·23-s − 278.·24-s − 1/5·25-s + 5.49·26-s + 1.05e3·27-s − 4.53·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{31} \cdot 17^{31} \cdot 79^{31}\right)^{s/2} \, \Gamma_{\C}(s)^{31} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{31} \cdot 17^{31} \cdot 79^{31}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{31} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 $$d$$ = $$62$$ $$N$$ = $$3^{31} \cdot 17^{31} \cdot 79^{31}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : induced by $\chi_{4029} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(62,\ 3^{31} \cdot 17^{31} \cdot 79^{31} ,\ ( \ : [1/2]^{31} ),\ 1 )$ $L(1)$ $\approx$ $1.440912861e10$ $L(\frac12)$ $\approx$ $1.440912861e10$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{3,\;17,\;79\}$,$$F_p(T)$$ is a polynomial of degree 62. If $p \in \{3,\;17,\;79\}$, then $F_p(T)$ is a polynomial of degree at most 61.
$p$$F_p(T)$
bad3 $$( 1 - T )^{31}$$
17 $$( 1 - T )^{31}$$
79 $$( 1 - T )^{31}$$
good2 $$1 - p^{2} T + 11 p T^{2} - 17 p^{2} T^{3} + 233 T^{4} - 19 p^{5} T^{5} + 1655 T^{6} - 959 p^{2} T^{7} + 4533 p T^{8} - 19259 T^{9} + 41297 T^{10} - 41021 p T^{11} + 163637 T^{12} - 308081 T^{13} + 580315 T^{14} - 1044633 T^{15} + 469181 p^{2} T^{16} - 3249959 T^{17} + 5606737 T^{18} - 9381695 T^{19} + 15618143 T^{20} - 25333337 T^{21} + 20420197 p T^{22} - 32182373 p T^{23} + 25184425 p^{2} T^{24} - 38625643 p^{2} T^{25} + 58794555 p^{2} T^{26} - 175690355 p T^{27} + 260412199 p T^{28} - 379272639 p T^{29} + 547859469 p T^{30} - 194503427 p^{3} T^{31} + 547859469 p^{2} T^{32} - 379272639 p^{3} T^{33} + 260412199 p^{4} T^{34} - 175690355 p^{5} T^{35} + 58794555 p^{7} T^{36} - 38625643 p^{8} T^{37} + 25184425 p^{9} T^{38} - 32182373 p^{9} T^{39} + 20420197 p^{10} T^{40} - 25333337 p^{10} T^{41} + 15618143 p^{11} T^{42} - 9381695 p^{12} T^{43} + 5606737 p^{13} T^{44} - 3249959 p^{14} T^{45} + 469181 p^{17} T^{46} - 1044633 p^{16} T^{47} + 580315 p^{17} T^{48} - 308081 p^{18} T^{49} + 163637 p^{19} T^{50} - 41021 p^{21} T^{51} + 41297 p^{21} T^{52} - 19259 p^{22} T^{53} + 4533 p^{24} T^{54} - 959 p^{26} T^{55} + 1655 p^{25} T^{56} - 19 p^{31} T^{57} + 233 p^{27} T^{58} - 17 p^{30} T^{59} + 11 p^{30} T^{60} - p^{32} T^{61} + p^{31} T^{62}$$
5 $$1 - 11 T + 122 T^{2} - 883 T^{3} + 6089 T^{4} - 34194 T^{5} + 182582 T^{6} - 855796 T^{7} + 3833972 T^{8} - 15620886 T^{9} + 61195564 T^{10} - 222437238 T^{11} + 781857304 T^{12} - 2582837269 T^{13} + 8295854922 T^{14} - 25271219961 T^{15} + 3010029937 p^{2} T^{16} - 214008019079 T^{17} + 598138392668 T^{18} - 1605685500532 T^{19} + 4258816407426 T^{20} - 10899556699048 T^{21} + 27701789760154 T^{22} - 68184312775503 T^{23} + 167424603459452 T^{24} - 79844076987537 p T^{25} + 953258757453772 T^{26} - 2214561180391339 T^{27} + 5166738076277903 T^{28} - 11742069586021617 T^{29} + 26848190966565363 T^{30} - 59833224778518096 T^{31} + 26848190966565363 p T^{32} - 11742069586021617 p^{2} T^{33} + 5166738076277903 p^{3} T^{34} - 2214561180391339 p^{4} T^{35} + 953258757453772 p^{5} T^{36} - 79844076987537 p^{7} T^{37} + 167424603459452 p^{7} T^{38} - 68184312775503 p^{8} T^{39} + 27701789760154 p^{9} T^{40} - 10899556699048 p^{10} T^{41} + 4258816407426 p^{11} T^{42} - 1605685500532 p^{12} T^{43} + 598138392668 p^{13} T^{44} - 214008019079 p^{14} T^{45} + 3010029937 p^{17} T^{46} - 25271219961 p^{16} T^{47} + 8295854922 p^{17} T^{48} - 2582837269 p^{18} T^{49} + 781857304 p^{19} T^{50} - 222437238 p^{20} T^{51} + 61195564 p^{21} T^{52} - 15620886 p^{22} T^{53} + 3833972 p^{23} T^{54} - 855796 p^{24} T^{55} + 182582 p^{25} T^{56} - 34194 p^{26} T^{57} + 6089 p^{27} T^{58} - 883 p^{28} T^{59} + 122 p^{29} T^{60} - 11 p^{30} T^{61} + p^{31} T^{62}$$
7 $$1 - 4 T + 101 T^{2} - 360 T^{3} + 5065 T^{4} - 16195 T^{5} + 168214 T^{6} - 483790 T^{7} + 4160493 T^{8} - 10752289 T^{9} + 81713764 T^{10} - 188901607 T^{11} + 1327892561 T^{12} - 2723680692 T^{13} + 2627397940 p T^{14} - 33082951722 T^{15} + 31763177220 p T^{16} - 49379101540 p T^{17} + 2395991608399 T^{18} - 3169145485116 T^{19} + 23470653216247 T^{20} - 26086128786106 T^{21} + 212799210553042 T^{22} - 198254372161166 T^{23} + 1812653558797101 T^{24} - 1436029574797828 T^{25} + 2093196102603086 p T^{26} - 10191643039679165 T^{27} + 112895966992408249 T^{28} - 71898595642639002 T^{29} + 829465698507864955 T^{30} - 504869317457408740 T^{31} + 829465698507864955 p T^{32} - 71898595642639002 p^{2} T^{33} + 112895966992408249 p^{3} T^{34} - 10191643039679165 p^{4} T^{35} + 2093196102603086 p^{6} T^{36} - 1436029574797828 p^{6} T^{37} + 1812653558797101 p^{7} T^{38} - 198254372161166 p^{8} T^{39} + 212799210553042 p^{9} T^{40} - 26086128786106 p^{10} T^{41} + 23470653216247 p^{11} T^{42} - 3169145485116 p^{12} T^{43} + 2395991608399 p^{13} T^{44} - 49379101540 p^{15} T^{45} + 31763177220 p^{16} T^{46} - 33082951722 p^{16} T^{47} + 2627397940 p^{18} T^{48} - 2723680692 p^{18} T^{49} + 1327892561 p^{19} T^{50} - 188901607 p^{20} T^{51} + 81713764 p^{21} T^{52} - 10752289 p^{22} T^{53} + 4160493 p^{23} T^{54} - 483790 p^{24} T^{55} + 168214 p^{25} T^{56} - 16195 p^{26} T^{57} + 5065 p^{27} T^{58} - 360 p^{28} T^{59} + 101 p^{29} T^{60} - 4 p^{30} T^{61} + p^{31} T^{62}$$
11 $$1 - 26 T + 45 p T^{2} - 6923 T^{3} + 82590 T^{4} - 846702 T^{5} + 7817393 T^{6} - 65389013 T^{7} + 506289717 T^{8} - 3644338347 T^{9} + 24681185527 T^{10} - 157759621991 T^{11} + 958891406895 T^{12} - 5554913037865 T^{13} + 30826866658481 T^{14} - 164155800062541 T^{15} + 841855923846998 T^{16} - 4163061095648030 T^{17} + 19904608878665578 T^{18} - 92098578205642951 T^{19} + 413240652338401612 T^{20} - 1799206511807962970 T^{21} + 7613376573838629262 T^{22} - 31323790749404730235 T^{23} +$$$$12\!\cdots\!98$$$$T^{24} -$$$$48\!\cdots\!17$$$$T^{25} +$$$$18\!\cdots\!75$$$$T^{26} -$$$$68\!\cdots\!67$$$$T^{27} +$$$$24\!\cdots\!26$$$$T^{28} -$$$$87\!\cdots\!31$$$$T^{29} +$$$$30\!\cdots\!58$$$$T^{30} -$$$$10\!\cdots\!98$$$$T^{31} +$$$$30\!\cdots\!58$$$$p T^{32} -$$$$87\!\cdots\!31$$$$p^{2} T^{33} +$$$$24\!\cdots\!26$$$$p^{3} T^{34} -$$$$68\!\cdots\!67$$$$p^{4} T^{35} +$$$$18\!\cdots\!75$$$$p^{5} T^{36} -$$$$48\!\cdots\!17$$$$p^{6} T^{37} +$$$$12\!\cdots\!98$$$$p^{7} T^{38} - 31323790749404730235 p^{8} T^{39} + 7613376573838629262 p^{9} T^{40} - 1799206511807962970 p^{10} T^{41} + 413240652338401612 p^{11} T^{42} - 92098578205642951 p^{12} T^{43} + 19904608878665578 p^{13} T^{44} - 4163061095648030 p^{14} T^{45} + 841855923846998 p^{15} T^{46} - 164155800062541 p^{16} T^{47} + 30826866658481 p^{17} T^{48} - 5554913037865 p^{18} T^{49} + 958891406895 p^{19} T^{50} - 157759621991 p^{20} T^{51} + 24681185527 p^{21} T^{52} - 3644338347 p^{22} T^{53} + 506289717 p^{23} T^{54} - 65389013 p^{24} T^{55} + 7817393 p^{25} T^{56} - 846702 p^{26} T^{57} + 82590 p^{27} T^{58} - 6923 p^{28} T^{59} + 45 p^{30} T^{60} - 26 p^{30} T^{61} + p^{31} T^{62}$$
13 $$1 - 7 T + 217 T^{2} - 1438 T^{3} + 23281 T^{4} - 145638 T^{5} + 1644127 T^{6} - 9700050 T^{7} + 85932630 T^{8} - 478321178 T^{9} + 3546538082 T^{10} - 18649114402 T^{11} + 120525524046 T^{12} - 599960931857 T^{13} + 3475971640098 T^{14} - 16424704821672 T^{15} + 87095214029705 T^{16} - 391958566139279 T^{17} + 1933045011167237 T^{18} - 8317025310572120 T^{19} + 38633400397693521 T^{20} - 159561336745858261 T^{21} + 705019963286908785 T^{22} - 2806056040726646268 T^{23} + 11881269375614170642 T^{24} - 45720505898254423633 T^{25} +$$$$18\!\cdots\!32$$$$T^{26} -$$$$69\!\cdots\!40$$$$T^{27} +$$$$27\!\cdots\!52$$$$T^{28} -$$$$99\!\cdots\!19$$$$T^{29} +$$$$37\!\cdots\!36$$$$T^{30} -$$$$13\!\cdots\!24$$$$T^{31} +$$$$37\!\cdots\!36$$$$p T^{32} -$$$$99\!\cdots\!19$$$$p^{2} T^{33} +$$$$27\!\cdots\!52$$$$p^{3} T^{34} -$$$$69\!\cdots\!40$$$$p^{4} T^{35} +$$$$18\!\cdots\!32$$$$p^{5} T^{36} - 45720505898254423633 p^{6} T^{37} + 11881269375614170642 p^{7} T^{38} - 2806056040726646268 p^{8} T^{39} + 705019963286908785 p^{9} T^{40} - 159561336745858261 p^{10} T^{41} + 38633400397693521 p^{11} T^{42} - 8317025310572120 p^{12} T^{43} + 1933045011167237 p^{13} T^{44} - 391958566139279 p^{14} T^{45} + 87095214029705 p^{15} T^{46} - 16424704821672 p^{16} T^{47} + 3475971640098 p^{17} T^{48} - 599960931857 p^{18} T^{49} + 120525524046 p^{19} T^{50} - 18649114402 p^{20} T^{51} + 3546538082 p^{21} T^{52} - 478321178 p^{22} T^{53} + 85932630 p^{23} T^{54} - 9700050 p^{24} T^{55} + 1644127 p^{25} T^{56} - 145638 p^{26} T^{57} + 23281 p^{27} T^{58} - 1438 p^{28} T^{59} + 217 p^{29} T^{60} - 7 p^{30} T^{61} + p^{31} T^{62}$$
19 $$1 - 32 T + 797 T^{2} - 14434 T^{3} + 224890 T^{4} - 2998090 T^{5} + 35965063 T^{6} - 389406765 T^{7} + 3890089045 T^{8} - 35998689827 T^{9} + 312286004951 T^{10} - 2548816918023 T^{11} + 19718711893334 T^{12} - 145053095119630 T^{13} + 53674505806681 p T^{14} - 6870824023278358 T^{15} + 44528740247189758 T^{16} - 278212205817607544 T^{17} + 1680645268651222257 T^{18} - 9833925462669299964 T^{19} + 55859413896932758183 T^{20} -$$$$30\!\cdots\!66$$$$T^{21} +$$$$16\!\cdots\!12$$$$T^{22} -$$$$86\!\cdots\!04$$$$T^{23} +$$$$44\!\cdots\!68$$$$T^{24} -$$$$22\!\cdots\!78$$$$T^{25} +$$$$10\!\cdots\!67$$$$T^{26} -$$$$52\!\cdots\!25$$$$T^{27} +$$$$12\!\cdots\!77$$$$p T^{28} -$$$$11\!\cdots\!13$$$$T^{29} +$$$$50\!\cdots\!44$$$$T^{30} -$$$$11\!\cdots\!62$$$$p T^{31} +$$$$50\!\cdots\!44$$$$p T^{32} -$$$$11\!\cdots\!13$$$$p^{2} T^{33} +$$$$12\!\cdots\!77$$$$p^{4} T^{34} -$$$$52\!\cdots\!25$$$$p^{4} T^{35} +$$$$10\!\cdots\!67$$$$p^{5} T^{36} -$$$$22\!\cdots\!78$$$$p^{6} T^{37} +$$$$44\!\cdots\!68$$$$p^{7} T^{38} -$$$$86\!\cdots\!04$$$$p^{8} T^{39} +$$$$16\!\cdots\!12$$$$p^{9} T^{40} -$$$$30\!\cdots\!66$$$$p^{10} T^{41} + 55859413896932758183 p^{11} T^{42} - 9833925462669299964 p^{12} T^{43} + 1680645268651222257 p^{13} T^{44} - 278212205817607544 p^{14} T^{45} + 44528740247189758 p^{15} T^{46} - 6870824023278358 p^{16} T^{47} + 53674505806681 p^{18} T^{48} - 145053095119630 p^{18} T^{49} + 19718711893334 p^{19} T^{50} - 2548816918023 p^{20} T^{51} + 312286004951 p^{21} T^{52} - 35998689827 p^{22} T^{53} + 3890089045 p^{23} T^{54} - 389406765 p^{24} T^{55} + 35965063 p^{25} T^{56} - 2998090 p^{26} T^{57} + 224890 p^{27} T^{58} - 14434 p^{28} T^{59} + 797 p^{29} T^{60} - 32 p^{30} T^{61} + p^{31} T^{62}$$
23 $$1 - 29 T + 789 T^{2} - 14354 T^{3} + 240958 T^{4} - 3328455 T^{5} + 42913459 T^{6} - 488499179 T^{7} + 5249703515 T^{8} - 51528104505 T^{9} + 481927085895 T^{10} - 4198657725709 T^{11} + 35117278316940 T^{12} - 277163697649189 T^{13} + 2113051203031493 T^{14} - 15341926845664816 T^{15} + 108151541336892196 T^{16} - 731059094344196297 T^{17} + 4818226240956978981 T^{18} - 30607824941396710960 T^{19} +$$$$19\!\cdots\!97$$$$T^{20} -$$$$11\!\cdots\!69$$$$T^{21} +$$$$67\!\cdots\!36$$$$T^{22} -$$$$38\!\cdots\!72$$$$T^{23} +$$$$21\!\cdots\!14$$$$T^{24} -$$$$11\!\cdots\!10$$$$T^{25} +$$$$63\!\cdots\!11$$$$T^{26} -$$$$33\!\cdots\!91$$$$T^{27} +$$$$17\!\cdots\!27$$$$T^{28} -$$$$86\!\cdots\!38$$$$T^{29} +$$$$18\!\cdots\!08$$$$p T^{30} -$$$$20\!\cdots\!58$$$$T^{31} +$$$$18\!\cdots\!08$$$$p^{2} T^{32} -$$$$86\!\cdots\!38$$$$p^{2} T^{33} +$$$$17\!\cdots\!27$$$$p^{3} T^{34} -$$$$33\!\cdots\!91$$$$p^{4} T^{35} +$$$$63\!\cdots\!11$$$$p^{5} T^{36} -$$$$11\!\cdots\!10$$$$p^{6} T^{37} +$$$$21\!\cdots\!14$$$$p^{7} T^{38} -$$$$38\!\cdots\!72$$$$p^{8} T^{39} +$$$$67\!\cdots\!36$$$$p^{9} T^{40} -$$$$11\!\cdots\!69$$$$p^{10} T^{41} +$$$$19\!\cdots\!97$$$$p^{11} T^{42} - 30607824941396710960 p^{12} T^{43} + 4818226240956978981 p^{13} T^{44} - 731059094344196297 p^{14} T^{45} + 108151541336892196 p^{15} T^{46} - 15341926845664816 p^{16} T^{47} + 2113051203031493 p^{17} T^{48} - 277163697649189 p^{18} T^{49} + 35117278316940 p^{19} T^{50} - 4198657725709 p^{20} T^{51} + 481927085895 p^{21} T^{52} - 51528104505 p^{22} T^{53} + 5249703515 p^{23} T^{54} - 488499179 p^{24} T^{55} + 42913459 p^{25} T^{56} - 3328455 p^{26} T^{57} + 240958 p^{27} T^{58} - 14354 p^{28} T^{59} + 789 p^{29} T^{60} - 29 p^{30} T^{61} + p^{31} T^{62}$$
29 $$1 - 25 T + 838 T^{2} - 15191 T^{3} + 302471 T^{4} - 4381756 T^{5} + 65796226 T^{6} - 802165204 T^{7} + 9919798799 T^{8} - 105042326449 T^{9} + 1118688911094 T^{10} - 10505128786787 T^{11} + 98953326169952 T^{12} - 835901580351844 T^{13} + 7086542845366729 T^{14} - 54383462069689070 T^{15} + 419955194647594555 T^{16} - 2946774293627915990 T^{17} + 20903837653180504336 T^{18} -$$$$13\!\cdots\!81$$$$T^{19} +$$$$88\!\cdots\!72$$$$T^{20} -$$$$52\!\cdots\!61$$$$T^{21} +$$$$31\!\cdots\!71$$$$T^{22} -$$$$17\!\cdots\!36$$$$T^{23} +$$$$98\!\cdots\!52$$$$T^{24} -$$$$49\!\cdots\!18$$$$T^{25} +$$$$26\!\cdots\!35$$$$T^{26} -$$$$12\!\cdots\!41$$$$T^{27} +$$$$65\!\cdots\!50$$$$T^{28} -$$$$28\!\cdots\!13$$$$T^{29} +$$$$16\!\cdots\!87$$$$T^{30} -$$$$75\!\cdots\!60$$$$T^{31} +$$$$16\!\cdots\!87$$$$p T^{32} -$$$$28\!\cdots\!13$$$$p^{2} T^{33} +$$$$65\!\cdots\!50$$$$p^{3} T^{34} -$$$$12\!\cdots\!41$$$$p^{4} T^{35} +$$$$26\!\cdots\!35$$$$p^{5} T^{36} -$$$$49\!\cdots\!18$$$$p^{6} T^{37} +$$$$98\!\cdots\!52$$$$p^{7} T^{38} -$$$$17\!\cdots\!36$$$$p^{8} T^{39} +$$$$31\!\cdots\!71$$$$p^{9} T^{40} -$$$$52\!\cdots\!61$$$$p^{10} T^{41} +$$$$88\!\cdots\!72$$$$p^{11} T^{42} -$$$$13\!\cdots\!81$$$$p^{12} T^{43} + 20903837653180504336 p^{13} T^{44} - 2946774293627915990 p^{14} T^{45} + 419955194647594555 p^{15} T^{46} - 54383462069689070 p^{16} T^{47} + 7086542845366729 p^{17} T^{48} - 835901580351844 p^{18} T^{49} + 98953326169952 p^{19} T^{50} - 10505128786787 p^{20} T^{51} + 1118688911094 p^{21} T^{52} - 105042326449 p^{22} T^{53} + 9919798799 p^{23} T^{54} - 802165204 p^{24} T^{55} + 65796226 p^{25} T^{56} - 4381756 p^{26} T^{57} + 302471 p^{27} T^{58} - 15191 p^{28} T^{59} + 838 p^{29} T^{60} - 25 p^{30} T^{61} + p^{31} T^{62}$$
31 $$1 - 22 T + 692 T^{2} - 12119 T^{3} + 226913 T^{4} - 3311904 T^{5} + 47494060 T^{6} - 596528197 T^{7} + 7174713584 T^{8} - 79371487954 T^{9} + 836290318500 T^{10} - 8289970758368 T^{11} + 78386862733087 T^{12} - 705263433588150 T^{13} + 6073345511042893 T^{14} - 50077101551894150 T^{15} + 396556508538330353 T^{16} - 3018694851491185766 T^{17} + 22134815062344915322 T^{18} -$$$$15\!\cdots\!76$$$$T^{19} +$$$$10\!\cdots\!22$$$$T^{20} -$$$$70\!\cdots\!56$$$$T^{21} +$$$$45\!\cdots\!87$$$$T^{22} -$$$$27\!\cdots\!71$$$$T^{23} +$$$$16\!\cdots\!05$$$$T^{24} -$$$$99\!\cdots\!30$$$$T^{25} +$$$$57\!\cdots\!41$$$$T^{26} -$$$$32\!\cdots\!21$$$$T^{27} +$$$$18\!\cdots\!77$$$$T^{28} -$$$$10\!\cdots\!74$$$$T^{29} +$$$$55\!\cdots\!79$$$$T^{30} -$$$$31\!\cdots\!36$$$$T^{31} +$$$$55\!\cdots\!79$$$$p T^{32} -$$$$10\!\cdots\!74$$$$p^{2} T^{33} +$$$$18\!\cdots\!77$$$$p^{3} T^{34} -$$$$32\!\cdots\!21$$$$p^{4} T^{35} +$$$$57\!\cdots\!41$$$$p^{5} T^{36} -$$$$99\!\cdots\!30$$$$p^{6} T^{37} +$$$$16\!\cdots\!05$$$$p^{7} T^{38} -$$$$27\!\cdots\!71$$$$p^{8} T^{39} +$$$$45\!\cdots\!87$$$$p^{9} T^{40} -$$$$70\!\cdots\!56$$$$p^{10} T^{41} +$$$$10\!\cdots\!22$$$$p^{11} T^{42} -$$$$15\!\cdots\!76$$$$p^{12} T^{43} + 22134815062344915322 p^{13} T^{44} - 3018694851491185766 p^{14} T^{45} + 396556508538330353 p^{15} T^{46} - 50077101551894150 p^{16} T^{47} + 6073345511042893 p^{17} T^{48} - 705263433588150 p^{18} T^{49} + 78386862733087 p^{19} T^{50} - 8289970758368 p^{20} T^{51} + 836290318500 p^{21} T^{52} - 79371487954 p^{22} T^{53} + 7174713584 p^{23} T^{54} - 596528197 p^{24} T^{55} + 47494060 p^{25} T^{56} - 3311904 p^{26} T^{57} + 226913 p^{27} T^{58} - 12119 p^{28} T^{59} + 692 p^{29} T^{60} - 22 p^{30} T^{61} + p^{31} T^{62}$$
37 $$1 + 4 T + 560 T^{2} + 2931 T^{3} + 161086 T^{4} + 1009261 T^{5} + 31821043 T^{6} + 224181493 T^{7} + 4851310274 T^{8} + 36667934951 T^{9} + 606506437033 T^{10} + 4747337512921 T^{11} + 64413499073131 T^{12} + 508786543761513 T^{13} + 5941445978922658 T^{14} + 46503422201284051 T^{15} + 483062477550635910 T^{16} + 3700808990678889048 T^{17} + 34983466144977014344 T^{18} +$$$$26\!\cdots\!36$$$$T^{19} +$$$$22\!\cdots\!80$$$$T^{20} +$$$$16\!\cdots\!04$$$$T^{21} +$$$$13\!\cdots\!60$$$$T^{22} +$$$$92\!\cdots\!67$$$$T^{23} +$$$$71\!\cdots\!39$$$$T^{24} +$$$$47\!\cdots\!15$$$$T^{25} +$$$$34\!\cdots\!43$$$$T^{26} +$$$$22\!\cdots\!94$$$$T^{27} +$$$$15\!\cdots\!91$$$$T^{28} +$$$$93\!\cdots\!08$$$$T^{29} +$$$$61\!\cdots\!19$$$$T^{30} +$$$$36\!\cdots\!22$$$$T^{31} +$$$$61\!\cdots\!19$$$$p T^{32} +$$$$93\!\cdots\!08$$$$p^{2} T^{33} +$$$$15\!\cdots\!91$$$$p^{3} T^{34} +$$$$22\!\cdots\!94$$$$p^{4} T^{35} +$$$$34\!\cdots\!43$$$$p^{5} T^{36} +$$$$47\!\cdots\!15$$$$p^{6} T^{37} +$$$$71\!\cdots\!39$$$$p^{7} T^{38} +$$$$92\!\cdots\!67$$$$p^{8} T^{39} +$$$$13\!\cdots\!60$$$$p^{9} T^{40} +$$$$16\!\cdots\!04$$$$p^{10} T^{41} +$$$$22\!\cdots\!80$$$$p^{11} T^{42} +$$$$26\!\cdots\!36$$$$p^{12} T^{43} + 34983466144977014344 p^{13} T^{44} + 3700808990678889048 p^{14} T^{45} + 483062477550635910 p^{15} T^{46} + 46503422201284051 p^{16} T^{47} + 5941445978922658 p^{17} T^{48} + 508786543761513 p^{18} T^{49} + 64413499073131 p^{19} T^{50} + 4747337512921 p^{20} T^{51} + 606506437033 p^{21} T^{52} + 36667934951 p^{22} T^{53} + 4851310274 p^{23} T^{54} + 224181493 p^{24} T^{55} + 31821043 p^{25} T^{56} + 1009261 p^{26} T^{57} + 161086 p^{27} T^{58} + 2931 p^{28} T^{59} + 560 p^{29} T^{60} + 4 p^{30} T^{61} + p^{31} T^{62}$$
41 $$1 - 33 T + 1167 T^{2} - 25949 T^{3} + 562270 T^{4} - 9777831 T^{5} + 163228530 T^{6} - 2383233284 T^{7} + 33389677353 T^{8} - 426520107032 T^{9} + 5243309042937 T^{10} - 60180674497718 T^{11} + 667008558909510 T^{12} - 7006126791093951 T^{13} + 71285331727031020 T^{14} - 694276933343249852 T^{15} + 6567061350815090582 T^{16} - 59872939203449277001 T^{17} + 12957409696342252292 p T^{18} -$$$$45\!\cdots\!00$$$$T^{19} +$$$$38\!\cdots\!29$$$$T^{20} -$$$$31\!\cdots\!86$$$$T^{21} +$$$$24\!\cdots\!93$$$$T^{22} -$$$$19\!\cdots\!20$$$$T^{23} +$$$$14\!\cdots\!88$$$$T^{24} -$$$$10\!\cdots\!71$$$$T^{25} +$$$$77\!\cdots\!72$$$$T^{26} -$$$$54\!\cdots\!33$$$$T^{27} +$$$$37\!\cdots\!63$$$$T^{28} -$$$$25\!\cdots\!59$$$$T^{29} +$$$$16\!\cdots\!93$$$$T^{30} -$$$$10\!\cdots\!32$$$$T^{31} +$$$$16\!\cdots\!93$$$$p T^{32} -$$$$25\!\cdots\!59$$$$p^{2} T^{33} +$$$$37\!\cdots\!63$$$$p^{3} T^{34} -$$$$54\!\cdots\!33$$$$p^{4} T^{35} +$$$$77\!\cdots\!72$$$$p^{5} T^{36} -$$$$10\!\cdots\!71$$$$p^{6} T^{37} +$$$$14\!\cdots\!88$$$$p^{7} T^{38} -$$$$19\!\cdots\!20$$$$p^{8} T^{39} +$$$$24\!\cdots\!93$$$$p^{9} T^{40} -$$$$31\!\cdots\!86$$$$p^{10} T^{41} +$$$$38\!\cdots\!29$$$$p^{11} T^{42} -$$$$45\!\cdots\!00$$$$p^{12} T^{43} + 12957409696342252292 p^{14} T^{44} - 59872939203449277001 p^{14} T^{45} + 6567061350815090582 p^{15} T^{46} - 694276933343249852 p^{16} T^{47} + 71285331727031020 p^{17} T^{48} - 7006126791093951 p^{18} T^{49} + 667008558909510 p^{19} T^{50} - 60180674497718 p^{20} T^{51} + 5243309042937 p^{21} T^{52} - 426520107032 p^{22} T^{53} + 33389677353 p^{23} T^{54} - 2383233284 p^{24} T^{55} + 163228530 p^{25} T^{56} - 9777831 p^{26} T^{57} + 562270 p^{27} T^{58} - 25949 p^{28} T^{59} + 1167 p^{29} T^{60} - 33 p^{30} T^{61} + p^{31} T^{62}$$
43 $$1 - 6 T + 632 T^{2} - 3519 T^{3} + 196481 T^{4} - 1014975 T^{5} + 40175738 T^{6} - 192891379 T^{7} + 6102008190 T^{8} - 27348904568 T^{9} + 737716333865 T^{10} - 3109087782600 T^{11} + 74317821917732 T^{12} - 297474482163921 T^{13} + 6446932370456517 T^{14} - 24801373451715531 T^{15} + 493540423150456887 T^{16} - 1847447664902893712 T^{17} + 33970211823292665636 T^{18} -$$$$12\!\cdots\!12$$$$T^{19} +$$$$21\!\cdots\!41$$$$T^{20} -$$$$77\!\cdots\!55$$$$T^{21} +$$$$12\!\cdots\!84$$$$T^{22} -$$$$10\!\cdots\!79$$$$p T^{23} +$$$$66\!\cdots\!28$$$$T^{24} -$$$$24\!\cdots\!74$$$$T^{25} +$$$$33\!\cdots\!29$$$$T^{26} -$$$$12\!\cdots\!97$$$$T^{27} +$$$$16\!\cdots\!92$$$$T^{28} -$$$$58\!\cdots\!33$$$$T^{29} +$$$$72\!\cdots\!99$$$$T^{30} -$$$$25\!\cdots\!42$$$$T^{31} +$$$$72\!\cdots\!99$$$$p T^{32} -$$$$58\!\cdots\!33$$$$p^{2} T^{33} +$$$$16\!\cdots\!92$$$$p^{3} T^{34} -$$$$12\!\cdots\!97$$$$p^{4} T^{35} +$$$$33\!\cdots\!29$$$$p^{5} T^{36} -$$$$24\!\cdots\!74$$$$p^{6} T^{37} +$$$$66\!\cdots\!28$$$$p^{7} T^{38} -$$$$10\!\cdots\!79$$$$p^{9} T^{39} +$$$$12\!\cdots\!84$$$$p^{9} T^{40} -$$$$77\!\cdots\!55$$$$p^{10} T^{41} +$$$$21\!\cdots\!41$$$$p^{11} T^{42} -$$$$12\!\cdots\!12$$$$p^{12} T^{43} + 33970211823292665636 p^{13} T^{44} - 1847447664902893712 p^{14} T^{45} + 493540423150456887 p^{15} T^{46} - 24801373451715531 p^{16} T^{47} + 6446932370456517 p^{17} T^{48} - 297474482163921 p^{18} T^{49} + 74317821917732 p^{19} T^{50} - 3109087782600 p^{20} T^{51} + 737716333865 p^{21} T^{52} - 27348904568 p^{22} T^{53} + 6102008190 p^{23} T^{54} - 192891379 p^{24} T^{55} + 40175738 p^{25} T^{56} - 1014975 p^{26} T^{57} + 196481 p^{27} T^{58} - 3519 p^{28} T^{59} + 632 p^{29} T^{60} - 6 p^{30} T^{61} + p^{31} T^{62}$$
47 $$1 - 23 T + 20 p T^{2} - 16293 T^{3} + 392134 T^{4} - 5578968 T^{5} + 101743358 T^{6} - 1246512159 T^{7} + 18973111215 T^{8} - 206320427684 T^{9} + 2757353265001 T^{10} - 27164874942201 T^{11} + 328836573884848 T^{12} - 2978171665944463 T^{13} + 33349137208008790 T^{14} - 280700343103479219 T^{15} + 2952165827697667494 T^{16} - 23290801062233463800 T^{17} +$$$$23\!\cdots\!71$$$$T^{18} -$$$$17\!\cdots\!20$$$$T^{19} +$$$$16\!\cdots\!01$$$$T^{20} -$$$$11\!\cdots\!10$$$$T^{21} +$$$$10\!\cdots\!84$$$$T^{22} -$$$$73\!\cdots\!41$$$$T^{23} +$$$$65\!\cdots\!61$$$$T^{24} -$$$$42\!\cdots\!72$$$$T^{25} +$$$$36\!\cdots\!20$$$$T^{26} -$$$$22\!\cdots\!02$$$$T^{27} +$$$$19\!\cdots\!01$$$$T^{28} -$$$$11\!\cdots\!56$$$$T^{29} +$$$$97\!\cdots\!55$$$$T^{30} -$$$$56\!\cdots\!94$$$$T^{31} +$$$$97\!\cdots\!55$$$$p T^{32} -$$$$11\!\cdots\!56$$$$p^{2} T^{33} +$$$$19\!\cdots\!01$$$$p^{3} T^{34} -$$$$22\!\cdots\!02$$$$p^{4} T^{35} +$$$$36\!\cdots\!20$$$$p^{5} T^{36} -$$$$42\!\cdots\!72$$$$p^{6} T^{37} +$$$$65\!\cdots\!61$$$$p^{7} T^{38} -$$$$73\!\cdots\!41$$$$p^{8} T^{39} +$$$$10\!\cdots\!84$$$$p^{9} T^{40} -$$$$11\!\cdots\!10$$$$p^{10} T^{41} +$$$$16\!\cdots\!01$$$$p^{11} T^{42} -$$$$17\!\cdots\!20$$$$p^{12} T^{43} +$$$$23\!\cdots\!71$$$$p^{13} T^{44} - 23290801062233463800 p^{14} T^{45} + 2952165827697667494 p^{15} T^{46} - 280700343103479219 p^{16} T^{47} + 33349137208008790 p^{17} T^{48} - 2978171665944463 p^{18} T^{49} + 328836573884848 p^{19} T^{50} - 27164874942201 p^{20} T^{51} + 2757353265001 p^{21} T^{52} - 206320427684 p^{22} T^{53} + 18973111215 p^{23} T^{54} - 1246512159 p^{24} T^{55} + 101743358 p^{25} T^{56} - 5578968 p^{26} T^{57} + 392134 p^{27} T^{58} - 16293 p^{28} T^{59} + 20 p^{30} T^{60} - 23 p^{30} T^{61} + p^{31} T^{62}$$
53 $$1 - 12 T + 947 T^{2} - 12435 T^{3} + 465392 T^{4} - 6302699 T^{5} + 157025297 T^{6} - 2105144474 T^{7} + 40466152584 T^{8} - 523863717511 T^{9} + 8409308820650 T^{10} - 103749111733293 T^{11} + 1456419536299473 T^{12} - 17022979339760056 T^{13} + 214988435045143040 T^{14} - 2376104666997792312 T^{15} + 518756296541567659 p T^{16} -$$$$28\!\cdots\!71$$$$T^{17} +$$$$30\!\cdots\!01$$$$T^{18} -$$$$30\!\cdots\!62$$$$T^{19} +$$$$30\!\cdots\!99$$$$T^{20} -$$$$28\!\cdots\!59$$$$T^{21} +$$$$27\!\cdots\!07$$$$T^{22} -$$$$24\!\cdots\!58$$$$T^{23} +$$$$21\!\cdots\!21$$$$T^{24} -$$$$34\!\cdots\!60$$$$p T^{25} +$$$$15\!\cdots\!86$$$$T^{26} -$$$$23\!\cdots\!04$$$$p T^{27} +$$$$99\!\cdots\!24$$$$T^{28} -$$$$76\!\cdots\!48$$$$T^{29} +$$$$58\!\cdots\!39$$$$T^{30} -$$$$42\!\cdots\!32$$$$T^{31} +$$$$58\!\cdots\!39$$$$p T^{32} -$$$$76\!\cdots\!48$$$$p^{2} T^{33} +$$$$99\!\cdots\!24$$$$p^{3} T^{34} -$$$$23\!\cdots\!04$$$$p^{5} T^{35} +$$$$15\!\cdots\!86$$$$p^{5} T^{36} -$$$$34\!\cdots\!60$$$$p^{7} T^{37} +$$$$21\!\cdots\!21$$$$p^{7} T^{38} -$$$$24\!\cdots\!58$$$$p^{8} T^{39} +$$$$27\!\cdots\!07$$$$p^{9} T^{40} -$$$$28\!\cdots\!59$$$$p^{10} T^{41} +$$$$30\!\cdots\!99$$$$p^{11} T^{42} -$$$$30\!\cdots\!62$$$$p^{12} T^{43} +$$$$30\!\cdots\!01$$$$p^{13} T^{44} -$$$$28\!\cdots\!71$$$$p^{14} T^{45} + 518756296541567659 p^{16} T^{46} - 2376104666997792312 p^{16} T^{47} + 214988435045143040 p^{17} T^{48} - 17022979339760056 p^{18} T^{49} + 1456419536299473 p^{19} T^{50} - 103749111733293 p^{20} T^{51} + 8409308820650 p^{21} T^{52} - 523863717511 p^{22} T^{53} + 40466152584 p^{23} T^{54} - 2105144474 p^{24} T^{55} + 157025297 p^{25} T^{56} - 6302699 p^{26} T^{57} + 465392 p^{27} T^{58} - 12435 p^{28} T^{59} + 947 p^{29} T^{60} - 12 p^{30} T^{61} + p^{31} T^{62}$$
59 $$1 - 27 T + 1242 T^{2} - 26160 T^{3} + 701120 T^{4} - 12286415 T^{5} + 247104339 T^{6} - 3747901670 T^{7} + 62265861524 T^{8} - 839863591718 T^{9} + 12121005281131 T^{10} - 148364499984971 T^{11} + 32531937486721 p T^{12} - 21663435927323393 T^{13} + 256753250584897716 T^{14} - 2707037884200110832 T^{15} + 29865066572246499300 T^{16} -$$$$29\!\cdots\!47$$$$T^{17} +$$$$30\!\cdots\!90$$$$T^{18} -$$$$29\!\cdots\!72$$$$T^{19} +$$$$28\!\cdots\!48$$$$T^{20} -$$$$26\!\cdots\!55$$$$T^{21} +$$$$24\!\cdots\!44$$$$T^{22} -$$$$21\!\cdots\!38$$$$T^{23} +$$$$19\!\cdots\!99$$$$T^{24} -$$$$16\!\cdots\!02$$$$T^{25} +$$$$14\!\cdots\!17$$$$T^{26} -$$$$11\!\cdots\!93$$$$T^{27} +$$$$96\!\cdots\!45$$$$T^{28} -$$$$75\!\cdots\!31$$$$T^{29} +$$$$60\!\cdots\!25$$$$T^{30} -$$$$45\!\cdots\!08$$$$T^{31} +$$$$60\!\cdots\!25$$$$p T^{32} -$$$$75\!\cdots\!31$$$$p^{2} T^{33} +$$$$96\!\cdots\!45$$$$p^{3} T^{34} -$$$$11\!\cdots\!93$$$$p^{4} T^{35} +$$$$14\!\cdots\!17$$$$p^{5} T^{36} -$$$$16\!\cdots\!02$$$$p^{6} T^{37} +$$$$19\!\cdots\!99$$$$p^{7} T^{38} -$$$$21\!\cdots\!38$$$$p^{8} T^{39} +$$$$24\!\cdots\!44$$$$p^{9} T^{40} -$$$$26\!\cdots\!55$$$$p^{10} T^{41} +$$$$28\!\cdots\!48$$$$p^{11} T^{42} -$$$$29\!\cdots\!72$$$$p^{12} T^{43} +$$$$30\!\cdots\!90$$$$p^{13} T^{44} -$$$$29\!\cdots\!47$$$$p^{14} T^{45} + 29865066572246499300 p^{15} T^{46} - 2707037884200110832 p^{16} T^{47} + 256753250584897716 p^{17} T^{48} - 21663435927323393 p^{18} T^{49} + 32531937486721 p^{20} T^{50} - 148364499984971 p^{20} T^{51} + 12121005281131 p^{21} T^{52} - 839863591718 p^{22} T^{53} + 62265861524 p^{23} T^{54} - 3747901670 p^{24} T^{55} + 247104339 p^{25} T^{56} - 12286415 p^{26} T^{57} + 701120 p^{27} T^{58} - 26160 p^{28} T^{59} + 1242 p^{29} T^{60} - 27 p^{30} T^{61} + p^{31} T^{62}$$
61 $$1 + 4 T + 1144 T^{2} + 4146 T^{3} + 645688 T^{4} + 2132146 T^{5} + 239873638 T^{6} + 725155869 T^{7} + 66018933570 T^{8} + 3006643637 p T^{9} + 14365380662229 T^{10} + 36772024084132 T^{11} + 2575648067619276 T^{12} + 6083955490271115 T^{13} + 391626509693210358 T^{14} + 853867743855430629 T^{15} + 51584310484234282978 T^{16} +$$$$10\!\cdots\!15$$$$T^{17} +$$$$59\!\cdots\!30$$$$T^{18} +$$$$11\!\cdots\!26$$$$T^{19} +$$$$61\!\cdots\!38$$$$T^{20} +$$$$10\!\cdots\!47$$$$T^{21} +$$$$57\!\cdots\!88$$$$T^{22} +$$$$90\!\cdots\!11$$$$T^{23} +$$$$49\!\cdots\!99$$$$T^{24} +$$$$71\!\cdots\!09$$$$T^{25} +$$$$38\!\cdots\!42$$$$T^{26} +$$$$51\!\cdots\!10$$$$T^{27} +$$$$27\!\cdots\!97$$$$T^{28} +$$$$56\!\cdots\!75$$$$p T^{29} +$$$$17\!\cdots\!40$$$$T^{30} +$$$$21\!\cdots\!94$$$$T^{31} +$$$$17\!\cdots\!40$$$$p T^{32} +$$$$56\!\cdots\!75$$$$p^{3} T^{33} +$$$$27\!\cdots\!97$$$$p^{3} T^{34} +$$$$51\!\cdots\!10$$$$p^{4} T^{35} +$$$$38\!\cdots\!42$$$$p^{5} T^{36} +$$$$71\!\cdots\!09$$$$p^{6} T^{37} +$$$$49\!\cdots\!99$$$$p^{7} T^{38} +$$$$90\!\cdots\!11$$$$p^{8} T^{39} +$$$$57\!\cdots\!88$$$$p^{9} T^{40} +$$$$10\!\cdots\!47$$$$p^{10} T^{41} +$$$$61\!\cdots\!38$$$$p^{11} T^{42} +$$$$11\!\cdots\!26$$$$p^{12} T^{43} +$$$$59\!\cdots\!30$$$$p^{13} T^{44} +$$$$10\!\cdots\!15$$$$p^{14} T^{45} + 51584310484234282978 p^{15} T^{46} + 853867743855430629 p^{16} T^{47} + 391626509693210358 p^{17} T^{48} + 6083955490271115 p^{18} T^{49} + 2575648067619276 p^{19} T^{50} + 36772024084132 p^{20} T^{51} + 14365380662229 p^{21} T^{52} + 3006643637 p^{23} T^{53} + 66018933570 p^{23} T^{54} + 725155869 p^{24} T^{55} + 239873638 p^{25} T^{56} + 2132146 p^{26} T^{57} + 645688 p^{27} T^{58} + 4146 p^{28} T^{59} + 1144 p^{29} T^{60} + 4 p^{30} T^{61} + p^{31} T^{62}$$
67 $$1 + 1152 T^{2} - 882 T^{3} + 666827 T^{4} - 954467 T^{5} + 258679776 T^{6} - 519652169 T^{7} + 75619146968 T^{8} - 189539431450 T^{9} + 17750420410623 T^{10} - 52012774851676 T^{11} + 3480500132319603 T^{12} - 11431063996972723 T^{13} + 585445881595245111 T^{14} - 2091186903360859184 T^{15} + 86087394796502625834 T^{16} -$$$$32\!\cdots\!59$$$$T^{17} +$$$$11\!\cdots\!20$$$$T^{18} -$$$$44\!\cdots\!36$$$$T^{19} +$$$$13\!\cdots\!78$$$$T^{20} -$$$$53\!\cdots\!20$$$$T^{21} +$$$$13\!\cdots\!71$$$$T^{22} -$$$$56\!\cdots\!79$$$$T^{23} +$$$$13\!\cdots\!75$$$$T^{24} -$$$$54\!\cdots\!31$$$$T^{25} +$$$$11\!\cdots\!10$$$$T^{26} -$$$$46\!\cdots\!82$$$$T^{27} +$$$$91\!\cdots\!77$$$$T^{28} -$$$$36\!\cdots\!54$$$$T^{29} +$$$$67\!\cdots\!64$$$$T^{30} -$$$$25\!\cdots\!20$$$$T^{31} +$$$$67\!\cdots\!64$$$$p T^{32} -$$$$36\!\cdots\!54$$$$p^{2} T^{33} +$$$$91\!\cdots\!77$$$$p^{3} T^{34} -$$$$46\!\cdots\!82$$$$p^{4} T^{35} +$$$$11\!\cdots\!10$$$$p^{5} T^{36} -$$$$54\!\cdots\!31$$$$p^{6} T^{37} +$$$$13\!\cdots\!75$$$$p^{7} T^{38} -$$$$56\!\cdots\!79$$$$p^{8} T^{39} +$$$$13\!\cdots\!71$$$$p^{9} T^{40} -$$$$53\!\cdots\!20$$$$p^{10} T^{41} +$$$$13\!\cdots\!78$$$$p^{11} T^{42} -$$$$44\!\cdots\!36$$$$p^{12} T^{43} +$$$$11\!\cdots\!20$$$$p^{13} T^{44} -$$$$32\!\cdots\!59$$$$p^{14} T^{45} + 86087394796502625834 p^{15} T^{46} - 2091186903360859184 p^{16} T^{47} + 585445881595245111 p^{17} T^{48} - 11431063996972723 p^{18} T^{49} + 3480500132319603 p^{19} T^{50} - 52012774851676 p^{20} T^{51} + 17750420410623 p^{21} T^{52} - 189539431450 p^{22} T^{53} + 75619146968 p^{23} T^{54} - 519652169 p^{24} T^{55} + 258679776 p^{25} T^{56} - 954467 p^{26} T^{57} + 666827 p^{27} T^{58} - 882 p^{28} T^{59} + 1152 p^{29} T^{60} + p^{31} T^{62}$$
71 $$1 - 35 T + 1789 T^{2} - 47874 T^{3} + 1463261 T^{4} - 32388909 T^{5} + 756142441 T^{6} - 14472480228 T^{7} + 282480583252 T^{8} - 4807930844668 T^{9} + 82114098683051 T^{10} - 1266656398133205 T^{11} + 19443856455980012 T^{12} - 275521914521326698 T^{13} + 3868030561073310990 T^{14} - 50852280306320434184 T^{15} +$$$$66\!\cdots\!49$$$$T^{16} -$$$$81\!\cdots\!96$$$$T^{17} +$$$$98\!\cdots\!19$$$$T^{18} -$$$$11\!\cdots\!85$$$$T^{19} +$$$$12\!\cdots\!88$$$$T^{20} -$$$$14\!\cdots\!40$$$$T^{21} +$$$$15\!\cdots\!41$$$$T^{22} -$$$$15\!\cdots\!20$$$$T^{23} +$$$$16\!\cdots\!46$$$$T^{24} -$$$$15\!\cdots\!45$$$$T^{25} +$$$$15\!\cdots\!30$$$$T^{26} -$$$$14\!\cdots\!56$$$$T^{27} +$$$$13\!\cdots\!08$$$$T^{28} -$$$$16\!\cdots\!15$$$$p T^{29} +$$$$10\!\cdots\!68$$$$T^{30} -$$$$87\!\cdots\!28$$$$T^{31} +$$$$10\!\cdots\!68$$$$p T^{32} -$$$$16\!\cdots\!15$$$$p^{3} T^{33} +$$$$13\!\cdots\!08$$$$p^{3} T^{34} -$$$$14\!\cdots\!56$$$$p^{4} T^{35} +$$$$15\!\cdots\!30$$$$p^{5} T^{36} -$$$$15\!\cdots\!45$$$$p^{6} T^{37} +$$$$16\!\cdots\!46$$$$p^{7} T^{38} -$$$$15\!\cdots\!20$$$$p^{8} T^{39} +$$$$15\!\cdots\!41$$$$p^{9} T^{40} -$$$$14\!\cdots\!40$$$$p^{10} T^{41} +$$$$12\!\cdots\!88$$$$p^{11} T^{42} -$$$$11\!\cdots\!85$$$$p^{12} T^{43} +$$$$98\!\cdots\!19$$$$p^{13} T^{44} -$$$$81\!\cdots\!96$$$$p^{14} T^{45} +$$$$66\!\cdots\!49$$$$p^{15} T^{46} - 50852280306320434184 p^{16} T^{47} + 3868030561073310990 p^{17} T^{48} - 275521914521326698 p^{18} T^{49} + 19443856455980012 p^{19} T^{50} - 1266656398133205 p^{20} T^{51} + 82114098683051 p^{21} T^{52} - 4807930844668 p^{22} T^{53} + 282480583252 p^{23} T^{54} - 14472480228 p^{24} T^{55} + 756142441 p^{25} T^{56} - 32388909 p^{26} T^{57} + 1463261 p^{27} T^{58} - 47874 p^{28} T^{59} + 1789 p^{29} T^{60} - 35 p^{30} T^{61} + p^{31} T^{62}$$
73 $$1 - 5 T + 1255 T^{2} - 8227 T^{3} + 784373 T^{4} - 6228953 T^{5} + 327198164 T^{6} - 2974522392 T^{7} + 102823184361 T^{8} - 1023394571536 T^{9} + 25996640753021 T^{10} - 273192165981362 T^{11} + 5504440224118539 T^{12} - 59320909173965448 T^{13} + 1001845270492856881 T^{14} - 10825560696230233369 T^{15} +$$$$15\!\cdots\!72$$$$T^{16} -$$$$17\!\cdots\!40$$$$T^{17} +$$$$22\!\cdots\!46$$$$T^{18} -$$$$23\!\cdots\!22$$$$T^{19} +$$$$28\!\cdots\!42$$$$T^{20} -$$$$28\!\cdots\!70$$$$T^{21} +$$$$32\!\cdots\!50$$$$T^{22} -$$$$31\!\cdots\!32$$$$T^{23} +$$$$33\!\cdots\!29$$$$T^{24} -$$$$31\!\cdots\!15$$$$T^{25} +$$$$31\!\cdots\!16$$$$T^{26} -$$$$28\!\cdots\!49$$$$T^{27} +$$$$27\!\cdots\!88$$$$T^{28} -$$$$23\!\cdots\!45$$$$T^{29} +$$$$21\!\cdots\!66$$$$T^{30} -$$$$17\!\cdots\!22$$$$T^{31} +$$$$21\!\cdots\!66$$$$p T^{32} -$$$$23\!\cdots\!45$$$$p^{2} T^{33} +$$$$27\!\cdots\!88$$$$p^{3} T^{34} -$$$$28\!\cdots\!49$$$$p^{4} T^{35} +$$$$31\!\cdots\!16$$$$p^{5} T^{36} -$$$$31\!\cdots\!15$$$$p^{6} T^{37} +$$$$33\!\cdots\!29$$$$p^{7} T^{38} -$$$$31\!\cdots\!32$$$$p^{8} T^{39} +$$$$32\!\cdots\!50$$$$p^{9} T^{40} -$$$$28\!\cdots\!70$$$$p^{10} T^{41} +$$$$28\!\cdots\!42$$$$p^{11} T^{42} -$$$$23\!\cdots\!22$$$$p^{12} T^{43} +$$$$22\!\cdots\!46$$$$p^{13} T^{44} -$$$$17\!\cdots\!40$$$$p^{14} T^{45} +$$$$15\!\cdots\!72$$$$p^{15} T^{46} - 10825560696230233369 p^{16} T^{47} + 1001845270492856881 p^{17} T^{48} - 59320909173965448 p^{18} T^{49} + 5504440224118539 p^{19} T^{50} - 273192165981362 p^{20} T^{51} + 25996640753021 p^{21} T^{52} - 1023394571536 p^{22} T^{53} + 102823184361 p^{23} T^{54} - 2974522392 p^{24} T^{55} + 327198164 p^{25} T^{56} - 6228953 p^{26} T^{57} + 784373 p^{27} T^{58} - 8227 p^{28} T^{59} + 1255 p^{29} T^{60} - 5 p^{30} T^{61} + p^{31} T^{62}$$
83 $$1 - 67 T + 3129 T^{2} - 108784 T^{3} + 3183423 T^{4} - 80458695 T^{5} + 1823170776 T^{6} - 37566942802 T^{7} + 716188737892 T^{8} - 12744009857243 T^{9} + 213742900678074 T^{10} - 3398602226957843 T^{11} + 51546558423455590 T^{12} - 748710605741306543 T^{13} + 10457533774287643927 T^{14} -$$$$14\!\cdots\!57$$$$T^{15} +$$$$18\!\cdots\!48$$$$T^{16} -$$$$23\!\cdots\!29$$$$T^{17} +$$$$28\!\cdots\!99$$$$T^{18} -$$$$33\!\cdots\!03$$$$T^{19} +$$$$39\!\cdots\!59$$$$T^{20} -$$$$44\!\cdots\!93$$$$T^{21} +$$$$49\!\cdots\!97$$$$T^{22} -$$$$53\!\cdots\!94$$$$T^{23} +$$$$56\!\cdots\!35$$$$T^{24} -$$$$58\!\cdots\!43$$$$T^{25} +$$$$59\!\cdots\!98$$$$T^{26} -$$$$58\!\cdots\!06$$$$T^{27} +$$$$57\!\cdots\!40$$$$T^{28} -$$$$54\!\cdots\!59$$$$T^{29} +$$$$51\!\cdots\!04$$$$T^{30} -$$$$47\!\cdots\!70$$$$T^{31} +$$$$51\!\cdots\!04$$$$p T^{32} -$$$$54\!\cdots\!59$$$$p^{2} T^{33} +$$$$57\!\cdots\!40$$$$p^{3} T^{34} -$$$$58\!\cdots\!06$$$$p^{4} T^{35} +$$$$59\!\cdots\!98$$$$p^{5} T^{36} -$$$$58\!\cdots\!43$$$$p^{6} T^{37} +$$$$56\!\cdots\!35$$$$p^{7} T^{38} -$$$$53\!\cdots\!94$$$$p^{8} T^{39} +$$$$49\!\cdots\!97$$$$p^{9} T^{40} -$$$$44\!\cdots\!93$$$$p^{10} T^{41} +$$$$39\!\cdots\!59$$$$p^{11} T^{42} -$$$$33\!\cdots\!03$$$$p^{12} T^{43} +$$$$28\!\cdots\!99$$$$p^{13} T^{44} -$$$$23\!\cdots\!29$$$$p^{14} T^{45} +$$$$18\!\cdots\!48$$$$p^{15} T^{46} -$$$$14\!\cdots\!57$$$$p^{16} T^{47} + 10457533774287643927 p^{17} T^{48} - 748710605741306543 p^{18} T^{49} + 51546558423455590 p^{19} T^{50} - 3398602226957843 p^{20} T^{51} + 213742900678074 p^{21} T^{52} - 12744009857243 p^{22} T^{53} + 716188737892 p^{23} T^{54} - 37566942802 p^{24} T^{55} + 1823170776 p^{25} T^{56} - 80458695 p^{26} T^{57} + 3183423 p^{27} T^{58} - 108784 p^{28} T^{59} + 3129 p^{29} T^{60} - 67 p^{30} T^{61} + p^{31} T^{62}$$
89 $$1 - 22 T + 1542 T^{2} - 29910 T^{3} + 1163971 T^{4} - 20373557 T^{5} + 578539469 T^{6} - 9295476649 T^{7} + 214290338046 T^{8} - 3201513000353 T^{9} + 63343254820268 T^{10} - 888687379951944 T^{11} + 15603470267896699 T^{12} - 207135416379044148 T^{13} + 3298837857605468744 T^{14} - 41676840208976983347 T^{15} +$$$$61\!\cdots\!93$$$$T^{16} -$$$$73\!\cdots\!36$$$$T^{17} +$$$$10\!\cdots\!64$$$$T^{18} -$$$$11\!\cdots\!84$$$$T^{19} +$$$$14\!\cdots\!41$$$$T^{20} -$$$$16\!\cdots\!35$$$$T^{21} +$$$$20\!\cdots\!33$$$$T^{22} -$$$$21\!\cdots\!55$$$$T^{23} +$$$$24\!\cdots\!28$$$$T^{24} -$$$$25\!\cdots\!35$$$$T^{25} +$$$$28\!\cdots\!76$$$$T^{26} -$$$$28\!\cdots\!10$$$$T^{27} +$$$$29\!\cdots\!87$$$$T^{28} -$$$$28\!\cdots\!94$$$$T^{29} +$$$$28\!\cdots\!02$$$$T^{30} -$$$$26\!\cdots\!42$$$$T^{31} +$$$$28\!\cdots\!02$$$$p T^{32} -$$$$28\!\cdots\!94$$$$p^{2} T^{33} +$$$$29\!\cdots\!87$$$$p^{3} T^{34} -$$$$28\!\cdots\!10$$$$p^{4} T^{35} +$$$$28\!\cdots\!76$$$$p^{5} T^{36} -$$$$25\!\cdots\!35$$$$p^{6} T^{37} +$$$$24\!\cdots\!28$$$$p^{7} T^{38} -$$$$21\!\cdots\!55$$$$p^{8} T^{39} +$$$$20\!\cdots\!33$$$$p^{9} T^{40} -$$$$16\!\cdots\!35$$$$p^{10} T^{41} +$$$$14\!\cdots\!41$$$$p^{11} T^{42} -$$$$11\!\cdots\!84$$$$p^{12} T^{43} +$$$$10\!\cdots\!64$$$$p^{13} T^{44} -$$$$73\!\cdots\!36$$$$p^{14} T^{45} +$$$$61\!\cdots\!93$$$$p^{15} T^{46} - 41676840208976983347 p^{16} T^{47} + 3298837857605468744 p^{17} T^{48} - 207135416379044148 p^{18} T^{49} + 15603470267896699 p^{19} T^{50} - 888687379951944 p^{20} T^{51} + 63343254820268 p^{21} T^{52} - 3201513000353 p^{22} T^{53} + 214290338046 p^{23} T^{54} - 9295476649 p^{24} T^{55} + 578539469 p^{25} T^{56} - 20373557 p^{26} T^{57} + 1163971 p^{27} T^{58} - 29910 p^{28} T^{59} + 1542 p^{29} T^{60} - 22 p^{30} T^{61} + p^{31} T^{62}$$
97 $$1 + 13 T + 2036 T^{2} + 25872 T^{3} + 2026752 T^{4} + 24977414 T^{5} + 1313426065 T^{6} + 15585206585 T^{7} + 622458798722 T^{8} + 7064444715895 T^{9} + 229782132210372 T^{10} + 2478612841271670 T^{11} + 68734901348987957 T^{12} + 700380849508040711 T^{13} + 17118684189800665551 T^{14} +$$$$16\!\cdots\!06$$$$T^{15} +$$$$36\!\cdots\!18$$$$T^{16} +$$$$32\!\cdots\!13$$$$T^{17} +$$$$66\!\cdots\!10$$$$T^{18} +$$$$54\!\cdots\!84$$$$T^{19} +$$$$10\!\cdots\!57$$$$T^{20} +$$$$80\!\cdots\!47$$$$T^{21} +$$$$14\!\cdots\!63$$$$T^{22} +$$$$10\!\cdots\!79$$$$T^{23} +$$$$19\!\cdots\!52$$$$T^{24} +$$$$12\!\cdots\!98$$$$T^{25} +$$$$22\!\cdots\!69$$$$T^{26} +$$$$13\!\cdots\!54$$$$T^{27} +$$$$24\!\cdots\!79$$$$T^{28} +$$$$13\!\cdots\!09$$$$T^{29} +$$$$24\!\cdots\!24$$$$T^{30} +$$$$12\!\cdots\!80$$$$T^{31} +$$$$24\!\cdots\!24$$$$p T^{32} +$$$$13\!\cdots\!09$$$$p^{2} T^{33} +$$$$24\!\cdots\!79$$$$p^{3} T^{34} +$$$$13\!\cdots\!54$$$$p^{4} T^{35} +$$$$22\!\cdots\!69$$$$p^{5} T^{36} +$$$$12\!\cdots\!98$$$$p^{6} T^{37} +$$$$19\!\cdots\!52$$$$p^{7} T^{38} +$$$$10\!\cdots\!79$$$$p^{8} T^{39} +$$$$14\!\cdots\!63$$$$p^{9} T^{40} +$$$$80\!\cdots\!47$$$$p^{10} T^{41} +$$$$10\!\cdots\!57$$$$p^{11} T^{42} +$$$$54\!\cdots\!84$$$$p^{12} T^{43} +$$$$66\!\cdots\!10$$$$p^{13} T^{44} +$$$$32\!\cdots\!13$$$$p^{14} T^{45} +$$$$36\!\cdots\!18$$$$p^{15} T^{46} +$$$$16\!\cdots\!06$$$$p^{16} T^{47} + 17118684189800665551 p^{17} T^{48} + 700380849508040711 p^{18} T^{49} + 68734901348987957 p^{19} T^{50} + 2478612841271670 p^{20} T^{51} + 229782132210372 p^{21} T^{52} + 7064444715895 p^{22} T^{53} + 622458798722 p^{23} T^{54} + 15585206585 p^{24} T^{55} + 1313426065 p^{25} T^{56} + 24977414 p^{26} T^{57} + 2026752 p^{27} T^{58} + 25872 p^{28} T^{59} + 2036 p^{29} T^{60} + 13 p^{30} T^{61} + p^{31} T^{62}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{62} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}