# Properties

 Degree 72 Conductor $2^{72} \cdot 3^{108}$ Sign $1$ Motivic weight 2 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 9·5-s + 3·9-s + 36·11-s − 18·23-s + 36·25-s − 18·29-s + 45·31-s − 198·41-s + 90·43-s − 27·45-s − 243·47-s + 36·49-s − 324·55-s + 252·59-s − 144·61-s + 108·67-s + 324·71-s − 63·73-s + 36·79-s − 9·81-s − 27·83-s − 567·89-s − 216·97-s + 108·99-s − 441·101-s + 72·103-s + 810·113-s + ⋯
 L(s)  = 1 − 9/5·5-s + 1/3·9-s + 3.27·11-s − 0.782·23-s + 1.43·25-s − 0.620·29-s + 1.45·31-s − 4.82·41-s + 2.09·43-s − 3/5·45-s − 5.17·47-s + 0.734·49-s − 5.89·55-s + 4.27·59-s − 2.36·61-s + 1.61·67-s + 4.56·71-s − 0.863·73-s + 0.455·79-s − 1/9·81-s − 0.325·83-s − 6.37·89-s − 2.22·97-s + 1.09·99-s − 4.36·101-s + 0.699·103-s + 7.16·113-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{108}\right)^{s/2} \, \Gamma_{\C}(s)^{36} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{108}\right)^{s/2} \, \Gamma_{\C}(s+1)^{36} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 $$d$$ = $$72$$ $$N$$ = $$2^{72} \cdot 3^{108}$$ $$\varepsilon$$ = $1$ motivic weight = $$2$$ character : induced by $\chi_{108} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$0$$ Selberg data = $$(72,\ 2^{72} \cdot 3^{108} ,\ ( \ : [1]^{36} ),\ 1 )$$ $$L(\frac{3}{2})$$ $$\approx$$ $$0.0660552$$ $$L(\frac12)$$ $$\approx$$ $$0.0660552$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3\}$,$$F_p(T)$$ is a polynomial of degree 72. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 71.
$p$$F_p(T)$
bad2 $$1$$
3 $$1 - p T^{2} + 2 p^{2} T^{4} + 31 p^{2} T^{5} - 188 p T^{6} - 79 p^{3} T^{7} + 5 p^{6} T^{8} + 212 p^{3} T^{9} + 94 p^{5} T^{10} - 643 p^{5} T^{11} - 4040 p^{4} T^{12} + 2731 p^{6} T^{13} - 197 p^{7} T^{14} - 514 p^{8} T^{15} - 3250 p^{8} T^{16} - 86 p^{12} T^{17} + 19252 p^{9} T^{18} - 86 p^{14} T^{19} - 3250 p^{12} T^{20} - 514 p^{14} T^{21} - 197 p^{15} T^{22} + 2731 p^{16} T^{23} - 4040 p^{16} T^{24} - 643 p^{19} T^{25} + 94 p^{21} T^{26} + 212 p^{21} T^{27} + 5 p^{26} T^{28} - 79 p^{25} T^{29} - 188 p^{25} T^{30} + 31 p^{28} T^{31} + 2 p^{30} T^{32} - p^{33} T^{34} + p^{36} T^{36}$$
good5 $$1 + 9 T + 9 p T^{2} - 189 T^{3} - 1611 T^{4} - 2853 p T^{5} - 29802 T^{6} - 182268 T^{7} + 463032 p T^{8} + 1916217 p T^{9} + 84343797 T^{10} - 121161753 T^{11} + 157699218 T^{12} - 15380876907 T^{13} - 17938368252 T^{14} - 270252182106 T^{15} + 1974731290983 T^{16} + 482876704926 T^{17} + 56056091685269 T^{18} - 155785217242572 T^{19} + 120695114619018 p T^{20} - 386387686503738 p^{2} T^{21} + 12589554738115827 T^{22} - 29927547775888587 p T^{23} + 1183208852096928927 T^{24} - 2182115467609779033 T^{25} + 32888209016721956931 T^{26} -$$$$10\!\cdots\!02$$$$T^{27} +$$$$38\!\cdots\!42$$$$T^{28} -$$$$49\!\cdots\!97$$$$T^{29} +$$$$13\!\cdots\!99$$$$T^{30} -$$$$79\!\cdots\!46$$$$T^{31} +$$$$49\!\cdots\!04$$$$T^{32} -$$$$36\!\cdots\!11$$$$p T^{33} +$$$$13\!\cdots\!44$$$$T^{34} -$$$$23\!\cdots\!06$$$$p^{2} T^{35} +$$$$22\!\cdots\!84$$$$T^{36} -$$$$23\!\cdots\!06$$$$p^{4} T^{37} +$$$$13\!\cdots\!44$$$$p^{4} T^{38} -$$$$36\!\cdots\!11$$$$p^{7} T^{39} +$$$$49\!\cdots\!04$$$$p^{8} T^{40} -$$$$79\!\cdots\!46$$$$p^{10} T^{41} +$$$$13\!\cdots\!99$$$$p^{12} T^{42} -$$$$49\!\cdots\!97$$$$p^{14} T^{43} +$$$$38\!\cdots\!42$$$$p^{16} T^{44} -$$$$10\!\cdots\!02$$$$p^{18} T^{45} + 32888209016721956931 p^{20} T^{46} - 2182115467609779033 p^{22} T^{47} + 1183208852096928927 p^{24} T^{48} - 29927547775888587 p^{27} T^{49} + 12589554738115827 p^{28} T^{50} - 386387686503738 p^{32} T^{51} + 120695114619018 p^{33} T^{52} - 155785217242572 p^{34} T^{53} + 56056091685269 p^{36} T^{54} + 482876704926 p^{38} T^{55} + 1974731290983 p^{40} T^{56} - 270252182106 p^{42} T^{57} - 17938368252 p^{44} T^{58} - 15380876907 p^{46} T^{59} + 157699218 p^{48} T^{60} - 121161753 p^{50} T^{61} + 84343797 p^{52} T^{62} + 1916217 p^{55} T^{63} + 463032 p^{57} T^{64} - 182268 p^{58} T^{65} - 29802 p^{60} T^{66} - 2853 p^{63} T^{67} - 1611 p^{64} T^{68} - 189 p^{66} T^{69} + 9 p^{69} T^{70} + 9 p^{70} T^{71} + p^{72} T^{72}$$
7 $$1 - 36 T^{2} + 120 T^{3} + 7281 T^{4} - 1152 p T^{5} - 269517 T^{6} - 304281 T^{7} + 3279987 p T^{8} + 4234708 T^{9} - 624361068 T^{10} - 5277792267 T^{11} + 26552323374 T^{12} + 314089728363 T^{13} + 1013731667658 T^{14} - 26120670738819 T^{15} - 85508050975632 T^{16} + 1165240443649743 T^{17} + 10257045338755108 T^{18} - 46198889207190495 T^{19} - 551169827134814988 T^{20} + 720618360823634445 T^{21} + 35853467263495563942 T^{22} + 47000089042961608629 T^{23} -$$$$15\!\cdots\!56$$$$T^{24} -$$$$83\!\cdots\!85$$$$T^{25} +$$$$58\!\cdots\!60$$$$T^{26} +$$$$42\!\cdots\!16$$$$T^{27} -$$$$12\!\cdots\!85$$$$T^{28} -$$$$31\!\cdots\!31$$$$T^{29} -$$$$55\!\cdots\!43$$$$p T^{30} +$$$$94\!\cdots\!28$$$$T^{31} +$$$$84\!\cdots\!15$$$$T^{32} -$$$$48\!\cdots\!10$$$$T^{33} -$$$$51\!\cdots\!04$$$$T^{34} +$$$$20\!\cdots\!04$$$$T^{35} +$$$$34\!\cdots\!26$$$$T^{36} +$$$$20\!\cdots\!04$$$$p^{2} T^{37} -$$$$51\!\cdots\!04$$$$p^{4} T^{38} -$$$$48\!\cdots\!10$$$$p^{6} T^{39} +$$$$84\!\cdots\!15$$$$p^{8} T^{40} +$$$$94\!\cdots\!28$$$$p^{10} T^{41} -$$$$55\!\cdots\!43$$$$p^{13} T^{42} -$$$$31\!\cdots\!31$$$$p^{14} T^{43} -$$$$12\!\cdots\!85$$$$p^{16} T^{44} +$$$$42\!\cdots\!16$$$$p^{18} T^{45} +$$$$58\!\cdots\!60$$$$p^{20} T^{46} -$$$$83\!\cdots\!85$$$$p^{22} T^{47} -$$$$15\!\cdots\!56$$$$p^{24} T^{48} + 47000089042961608629 p^{26} T^{49} + 35853467263495563942 p^{28} T^{50} + 720618360823634445 p^{30} T^{51} - 551169827134814988 p^{32} T^{52} - 46198889207190495 p^{34} T^{53} + 10257045338755108 p^{36} T^{54} + 1165240443649743 p^{38} T^{55} - 85508050975632 p^{40} T^{56} - 26120670738819 p^{42} T^{57} + 1013731667658 p^{44} T^{58} + 314089728363 p^{46} T^{59} + 26552323374 p^{48} T^{60} - 5277792267 p^{50} T^{61} - 624361068 p^{52} T^{62} + 4234708 p^{54} T^{63} + 3279987 p^{57} T^{64} - 304281 p^{58} T^{65} - 269517 p^{60} T^{66} - 1152 p^{63} T^{67} + 7281 p^{64} T^{68} + 120 p^{66} T^{69} - 36 p^{68} T^{70} + p^{72} T^{72}$$
11 $$1 - 36 T + 828 T^{2} - 12042 T^{3} + 127395 T^{4} - 884889 T^{5} + 3094368 T^{6} + 33980112 T^{7} - 860439384 T^{8} + 12028952385 T^{9} - 71429536467 T^{10} - 626725979523 T^{11} + 22860031230066 T^{12} - 225124859349 p^{3} T^{13} + 192603315816243 p T^{14} - 6223933646267220 T^{15} - 67455524379915000 T^{16} + 1184100180986861505 T^{17} - 15814028936542017865 T^{18} +$$$$16\!\cdots\!67$$$$T^{19} -$$$$95\!\cdots\!68$$$$T^{20} -$$$$11\!\cdots\!61$$$$p T^{21} +$$$$35\!\cdots\!32$$$$T^{22} -$$$$60\!\cdots\!09$$$$T^{23} +$$$$78\!\cdots\!50$$$$T^{24} -$$$$11\!\cdots\!38$$$$T^{25} +$$$$15\!\cdots\!37$$$$T^{26} -$$$$17\!\cdots\!39$$$$T^{27} +$$$$13\!\cdots\!12$$$$T^{28} -$$$$37\!\cdots\!01$$$$T^{29} -$$$$68\!\cdots\!50$$$$T^{30} +$$$$13\!\cdots\!85$$$$T^{31} -$$$$12\!\cdots\!00$$$$T^{32} +$$$$31\!\cdots\!23$$$$T^{33} +$$$$13\!\cdots\!78$$$$T^{34} -$$$$31\!\cdots\!30$$$$T^{35} +$$$$43\!\cdots\!12$$$$T^{36} -$$$$31\!\cdots\!30$$$$p^{2} T^{37} +$$$$13\!\cdots\!78$$$$p^{4} T^{38} +$$$$31\!\cdots\!23$$$$p^{6} T^{39} -$$$$12\!\cdots\!00$$$$p^{8} T^{40} +$$$$13\!\cdots\!85$$$$p^{10} T^{41} -$$$$68\!\cdots\!50$$$$p^{12} T^{42} -$$$$37\!\cdots\!01$$$$p^{14} T^{43} +$$$$13\!\cdots\!12$$$$p^{16} T^{44} -$$$$17\!\cdots\!39$$$$p^{18} T^{45} +$$$$15\!\cdots\!37$$$$p^{20} T^{46} -$$$$11\!\cdots\!38$$$$p^{22} T^{47} +$$$$78\!\cdots\!50$$$$p^{24} T^{48} -$$$$60\!\cdots\!09$$$$p^{26} T^{49} +$$$$35\!\cdots\!32$$$$p^{28} T^{50} -$$$$11\!\cdots\!61$$$$p^{31} T^{51} -$$$$95\!\cdots\!68$$$$p^{32} T^{52} +$$$$16\!\cdots\!67$$$$p^{34} T^{53} - 15814028936542017865 p^{36} T^{54} + 1184100180986861505 p^{38} T^{55} - 67455524379915000 p^{40} T^{56} - 6223933646267220 p^{42} T^{57} + 192603315816243 p^{45} T^{58} - 225124859349 p^{49} T^{59} + 22860031230066 p^{48} T^{60} - 626725979523 p^{50} T^{61} - 71429536467 p^{52} T^{62} + 12028952385 p^{54} T^{63} - 860439384 p^{56} T^{64} + 33980112 p^{58} T^{65} + 3094368 p^{60} T^{66} - 884889 p^{62} T^{67} + 127395 p^{64} T^{68} - 12042 p^{66} T^{69} + 828 p^{68} T^{70} - 36 p^{70} T^{71} + p^{72} T^{72}$$
13 $$1 - 9 p T^{2} - 123 T^{3} - 23607 T^{4} + 239013 T^{5} + 7444680 T^{6} + 136722744 T^{7} - 59887674 p T^{8} - 9424541132 T^{9} + 10197594306 T^{10} - 2741453733087 T^{11} + 55358667958128 T^{12} + 761094566245071 T^{13} + 9912519842633667 T^{14} - 88011399615903708 T^{15} + 464524114419535581 T^{16} + 1791730558886494833 T^{17} -$$$$21\!\cdots\!48$$$$T^{18} +$$$$63\!\cdots\!00$$$$T^{19} +$$$$35\!\cdots\!23$$$$T^{20} +$$$$51\!\cdots\!25$$$$T^{21} -$$$$12\!\cdots\!29$$$$T^{22} +$$$$18\!\cdots\!63$$$$T^{23} -$$$$40\!\cdots\!84$$$$T^{24} -$$$$11\!\cdots\!23$$$$T^{25} +$$$$31\!\cdots\!80$$$$p T^{26} -$$$$13\!\cdots\!99$$$$T^{27} +$$$$47\!\cdots\!66$$$$T^{28} +$$$$66\!\cdots\!25$$$$T^{29} +$$$$18\!\cdots\!84$$$$T^{30} -$$$$82\!\cdots\!12$$$$T^{31} +$$$$59\!\cdots\!71$$$$T^{32} +$$$$13\!\cdots\!72$$$$T^{33} -$$$$24\!\cdots\!13$$$$T^{34} +$$$$56\!\cdots\!66$$$$T^{35} +$$$$66\!\cdots\!62$$$$T^{36} +$$$$56\!\cdots\!66$$$$p^{2} T^{37} -$$$$24\!\cdots\!13$$$$p^{4} T^{38} +$$$$13\!\cdots\!72$$$$p^{6} T^{39} +$$$$59\!\cdots\!71$$$$p^{8} T^{40} -$$$$82\!\cdots\!12$$$$p^{10} T^{41} +$$$$18\!\cdots\!84$$$$p^{12} T^{42} +$$$$66\!\cdots\!25$$$$p^{14} T^{43} +$$$$47\!\cdots\!66$$$$p^{16} T^{44} -$$$$13\!\cdots\!99$$$$p^{18} T^{45} +$$$$31\!\cdots\!80$$$$p^{21} T^{46} -$$$$11\!\cdots\!23$$$$p^{22} T^{47} -$$$$40\!\cdots\!84$$$$p^{24} T^{48} +$$$$18\!\cdots\!63$$$$p^{26} T^{49} -$$$$12\!\cdots\!29$$$$p^{28} T^{50} +$$$$51\!\cdots\!25$$$$p^{30} T^{51} +$$$$35\!\cdots\!23$$$$p^{32} T^{52} +$$$$63\!\cdots\!00$$$$p^{34} T^{53} -$$$$21\!\cdots\!48$$$$p^{36} T^{54} + 1791730558886494833 p^{38} T^{55} + 464524114419535581 p^{40} T^{56} - 88011399615903708 p^{42} T^{57} + 9912519842633667 p^{44} T^{58} + 761094566245071 p^{46} T^{59} + 55358667958128 p^{48} T^{60} - 2741453733087 p^{50} T^{61} + 10197594306 p^{52} T^{62} - 9424541132 p^{54} T^{63} - 59887674 p^{57} T^{64} + 136722744 p^{58} T^{65} + 7444680 p^{60} T^{66} + 239013 p^{62} T^{67} - 23607 p^{64} T^{68} - 123 p^{66} T^{69} - 9 p^{69} T^{70} + p^{72} T^{72}$$
17 $$1 + 144 p T^{2} + 2887173 T^{4} - 5184 p^{2} T^{5} + 2172762780 T^{6} - 2488497228 T^{7} + 68718267198 p T^{8} - 84659364864 p T^{9} + 28173531458340 p T^{10} + 180677242526184 T^{11} + 157032946781972985 T^{12} + 890524092410925183 T^{13} + 43003698513828535368 T^{14} +$$$$75\!\cdots\!60$$$$T^{15} +$$$$10\!\cdots\!63$$$$T^{16} +$$$$40\!\cdots\!45$$$$T^{17} +$$$$28\!\cdots\!57$$$$T^{18} +$$$$15\!\cdots\!22$$$$T^{19} +$$$$11\!\cdots\!28$$$$T^{20} +$$$$49\!\cdots\!12$$$$T^{21} +$$$$56\!\cdots\!45$$$$T^{22} +$$$$12\!\cdots\!42$$$$T^{23} +$$$$26\!\cdots\!17$$$$T^{24} +$$$$29\!\cdots\!09$$$$T^{25} +$$$$10\!\cdots\!53$$$$T^{26} +$$$$78\!\cdots\!68$$$$T^{27} +$$$$35\!\cdots\!57$$$$T^{28} +$$$$28\!\cdots\!95$$$$T^{29} +$$$$97\!\cdots\!06$$$$T^{30} +$$$$12\!\cdots\!39$$$$T^{31} +$$$$22\!\cdots\!18$$$$T^{32} +$$$$28\!\cdots\!28$$$$p T^{33} +$$$$17\!\cdots\!13$$$$p^{2} T^{34} +$$$$34\!\cdots\!85$$$$p^{3} T^{35} +$$$$15\!\cdots\!12$$$$p^{4} T^{36} +$$$$34\!\cdots\!85$$$$p^{5} T^{37} +$$$$17\!\cdots\!13$$$$p^{6} T^{38} +$$$$28\!\cdots\!28$$$$p^{7} T^{39} +$$$$22\!\cdots\!18$$$$p^{8} T^{40} +$$$$12\!\cdots\!39$$$$p^{10} T^{41} +$$$$97\!\cdots\!06$$$$p^{12} T^{42} +$$$$28\!\cdots\!95$$$$p^{14} T^{43} +$$$$35\!\cdots\!57$$$$p^{16} T^{44} +$$$$78\!\cdots\!68$$$$p^{18} T^{45} +$$$$10\!\cdots\!53$$$$p^{20} T^{46} +$$$$29\!\cdots\!09$$$$p^{22} T^{47} +$$$$26\!\cdots\!17$$$$p^{24} T^{48} +$$$$12\!\cdots\!42$$$$p^{26} T^{49} +$$$$56\!\cdots\!45$$$$p^{28} T^{50} +$$$$49\!\cdots\!12$$$$p^{30} T^{51} +$$$$11\!\cdots\!28$$$$p^{32} T^{52} +$$$$15\!\cdots\!22$$$$p^{34} T^{53} +$$$$28\!\cdots\!57$$$$p^{36} T^{54} +$$$$40\!\cdots\!45$$$$p^{38} T^{55} +$$$$10\!\cdots\!63$$$$p^{40} T^{56} +$$$$75\!\cdots\!60$$$$p^{42} T^{57} + 43003698513828535368 p^{44} T^{58} + 890524092410925183 p^{46} T^{59} + 157032946781972985 p^{48} T^{60} + 180677242526184 p^{50} T^{61} + 28173531458340 p^{53} T^{62} - 84659364864 p^{55} T^{63} + 68718267198 p^{57} T^{64} - 2488497228 p^{58} T^{65} + 2172762780 p^{60} T^{66} - 5184 p^{64} T^{67} + 2887173 p^{64} T^{68} + 144 p^{69} T^{70} + p^{72} T^{72}$$
19 $$1 - 162 p T^{2} - 25458 T^{3} + 4631337 T^{4} + 73952946 T^{5} - 4153245213 T^{6} - 103964509395 T^{7} + 2205104051394 T^{8} + 89559936881398 T^{9} - 456201089207127 T^{10} - 49904834811929307 T^{11} - 245905923503181873 T^{12} + 16999747986533099589 T^{13} +$$$$21\!\cdots\!45$$$$T^{14} -$$$$13\!\cdots\!24$$$$p T^{15} -$$$$46\!\cdots\!03$$$$T^{16} +$$$$11\!\cdots\!75$$$$T^{17} -$$$$21\!\cdots\!61$$$$T^{18} -$$$$59\!\cdots\!03$$$$T^{19} +$$$$13\!\cdots\!27$$$$T^{20} +$$$$63\!\cdots\!20$$$$T^{21} +$$$$13\!\cdots\!28$$$$T^{22} -$$$$27\!\cdots\!35$$$$T^{23} -$$$$42\!\cdots\!23$$$$T^{24} +$$$$42\!\cdots\!82$$$$T^{25} +$$$$18\!\cdots\!19$$$$T^{26} +$$$$13\!\cdots\!34$$$$T^{27} -$$$$16\!\cdots\!88$$$$T^{28} -$$$$48\!\cdots\!06$$$$T^{29} -$$$$12\!\cdots\!71$$$$T^{30} -$$$$23\!\cdots\!32$$$$T^{31} +$$$$22\!\cdots\!14$$$$T^{32} +$$$$19\!\cdots\!06$$$$T^{33} +$$$$27\!\cdots\!84$$$$T^{34} -$$$$20\!\cdots\!94$$$$p T^{35} -$$$$25\!\cdots\!76$$$$p^{3} T^{36} -$$$$20\!\cdots\!94$$$$p^{3} T^{37} +$$$$27\!\cdots\!84$$$$p^{4} T^{38} +$$$$19\!\cdots\!06$$$$p^{6} T^{39} +$$$$22\!\cdots\!14$$$$p^{8} T^{40} -$$$$23\!\cdots\!32$$$$p^{10} T^{41} -$$$$12\!\cdots\!71$$$$p^{12} T^{42} -$$$$48\!\cdots\!06$$$$p^{14} T^{43} -$$$$16\!\cdots\!88$$$$p^{16} T^{44} +$$$$13\!\cdots\!34$$$$p^{18} T^{45} +$$$$18\!\cdots\!19$$$$p^{20} T^{46} +$$$$42\!\cdots\!82$$$$p^{22} T^{47} -$$$$42\!\cdots\!23$$$$p^{24} T^{48} -$$$$27\!\cdots\!35$$$$p^{26} T^{49} +$$$$13\!\cdots\!28$$$$p^{28} T^{50} +$$$$63\!\cdots\!20$$$$p^{30} T^{51} +$$$$13\!\cdots\!27$$$$p^{32} T^{52} -$$$$59\!\cdots\!03$$$$p^{34} T^{53} -$$$$21\!\cdots\!61$$$$p^{36} T^{54} +$$$$11\!\cdots\!75$$$$p^{38} T^{55} -$$$$46\!\cdots\!03$$$$p^{40} T^{56} -$$$$13\!\cdots\!24$$$$p^{43} T^{57} +$$$$21\!\cdots\!45$$$$p^{44} T^{58} + 16999747986533099589 p^{46} T^{59} - 245905923503181873 p^{48} T^{60} - 49904834811929307 p^{50} T^{61} - 456201089207127 p^{52} T^{62} + 89559936881398 p^{54} T^{63} + 2205104051394 p^{56} T^{64} - 103964509395 p^{58} T^{65} - 4153245213 p^{60} T^{66} + 73952946 p^{62} T^{67} + 4631337 p^{64} T^{68} - 25458 p^{66} T^{69} - 162 p^{69} T^{70} + p^{72} T^{72}$$
23 $$1 + 18 T - 2115 T^{2} - 39096 T^{3} + 2933721 T^{4} + 40981068 T^{5} - 3559438020 T^{6} - 29128843278 T^{7} + 3795197480973 T^{8} + 11493748091142 T^{9} - 3527515213800417 T^{10} + 7272373837068342 T^{11} + 2940058537517042139 T^{12} - 20410434166936941504 T^{13} -$$$$22\!\cdots\!36$$$$T^{14} +$$$$26\!\cdots\!47$$$$T^{15} +$$$$14\!\cdots\!66$$$$T^{16} -$$$$26\!\cdots\!01$$$$T^{17} -$$$$88\!\cdots\!57$$$$T^{18} +$$$$22\!\cdots\!23$$$$T^{19} +$$$$46\!\cdots\!51$$$$T^{20} -$$$$16\!\cdots\!08$$$$T^{21} -$$$$20\!\cdots\!13$$$$T^{22} +$$$$11\!\cdots\!75$$$$T^{23} +$$$$57\!\cdots\!83$$$$T^{24} -$$$$69\!\cdots\!55$$$$T^{25} +$$$$71\!\cdots\!52$$$$T^{26} +$$$$38\!\cdots\!73$$$$T^{27} -$$$$27\!\cdots\!28$$$$T^{28} -$$$$19\!\cdots\!00$$$$T^{29} +$$$$28\!\cdots\!82$$$$T^{30} +$$$$85\!\cdots\!18$$$$T^{31} -$$$$20\!\cdots\!08$$$$T^{32} -$$$$29\!\cdots\!89$$$$T^{33} +$$$$13\!\cdots\!78$$$$T^{34} +$$$$55\!\cdots\!41$$$$T^{35} -$$$$74\!\cdots\!88$$$$T^{36} +$$$$55\!\cdots\!41$$$$p^{2} T^{37} +$$$$13\!\cdots\!78$$$$p^{4} T^{38} -$$$$29\!\cdots\!89$$$$p^{6} T^{39} -$$$$20\!\cdots\!08$$$$p^{8} T^{40} +$$$$85\!\cdots\!18$$$$p^{10} T^{41} +$$$$28\!\cdots\!82$$$$p^{12} T^{42} -$$$$19\!\cdots\!00$$$$p^{14} T^{43} -$$$$27\!\cdots\!28$$$$p^{16} T^{44} +$$$$38\!\cdots\!73$$$$p^{18} T^{45} +$$$$71\!\cdots\!52$$$$p^{20} T^{46} -$$$$69\!\cdots\!55$$$$p^{22} T^{47} +$$$$57\!\cdots\!83$$$$p^{24} T^{48} +$$$$11\!\cdots\!75$$$$p^{26} T^{49} -$$$$20\!\cdots\!13$$$$p^{28} T^{50} -$$$$16\!\cdots\!08$$$$p^{30} T^{51} +$$$$46\!\cdots\!51$$$$p^{32} T^{52} +$$$$22\!\cdots\!23$$$$p^{34} T^{53} -$$$$88\!\cdots\!57$$$$p^{36} T^{54} -$$$$26\!\cdots\!01$$$$p^{38} T^{55} +$$$$14\!\cdots\!66$$$$p^{40} T^{56} +$$$$26\!\cdots\!47$$$$p^{42} T^{57} -$$$$22\!\cdots\!36$$$$p^{44} T^{58} - 20410434166936941504 p^{46} T^{59} + 2940058537517042139 p^{48} T^{60} + 7272373837068342 p^{50} T^{61} - 3527515213800417 p^{52} T^{62} + 11493748091142 p^{54} T^{63} + 3795197480973 p^{56} T^{64} - 29128843278 p^{58} T^{65} - 3559438020 p^{60} T^{66} + 40981068 p^{62} T^{67} + 2933721 p^{64} T^{68} - 39096 p^{66} T^{69} - 2115 p^{68} T^{70} + 18 p^{70} T^{71} + p^{72} T^{72}$$
29 $$1 + 18 T - 3843 T^{2} + 48465 T^{3} + 10962684 T^{4} - 352435905 T^{5} - 17197385286 T^{6} + 1124338521924 T^{7} + 9344152756674 T^{8} - 2246797049860179 T^{9} + 30972045151146096 T^{10} + 2997102683076161391 T^{11} -$$$$10\!\cdots\!52$$$$T^{12} -$$$$22\!\cdots\!72$$$$T^{13} +$$$$19\!\cdots\!96$$$$T^{14} -$$$$88\!\cdots\!91$$$$T^{15} -$$$$24\!\cdots\!07$$$$T^{16} +$$$$58\!\cdots\!95$$$$T^{17} +$$$$18\!\cdots\!02$$$$T^{18} -$$$$10\!\cdots\!49$$$$T^{19} -$$$$20\!\cdots\!91$$$$T^{20} +$$$$12\!\cdots\!49$$$$T^{21} -$$$$20\!\cdots\!74$$$$T^{22} -$$$$32\!\cdots\!38$$$$p T^{23} +$$$$38\!\cdots\!53$$$$T^{24} +$$$$25\!\cdots\!51$$$$T^{25} -$$$$42\!\cdots\!29$$$$T^{26} +$$$$53\!\cdots\!82$$$$T^{27} +$$$$31\!\cdots\!29$$$$T^{28} -$$$$10\!\cdots\!09$$$$T^{29} -$$$$10\!\cdots\!97$$$$T^{30} +$$$$11\!\cdots\!16$$$$T^{31} -$$$$11\!\cdots\!94$$$$T^{32} -$$$$82\!\cdots\!99$$$$T^{33} +$$$$25\!\cdots\!89$$$$T^{34} +$$$$27\!\cdots\!67$$$$T^{35} -$$$$26\!\cdots\!42$$$$T^{36} +$$$$27\!\cdots\!67$$$$p^{2} T^{37} +$$$$25\!\cdots\!89$$$$p^{4} T^{38} -$$$$82\!\cdots\!99$$$$p^{6} T^{39} -$$$$11\!\cdots\!94$$$$p^{8} T^{40} +$$$$11\!\cdots\!16$$$$p^{10} T^{41} -$$$$10\!\cdots\!97$$$$p^{12} T^{42} -$$$$10\!\cdots\!09$$$$p^{14} T^{43} +$$$$31\!\cdots\!29$$$$p^{16} T^{44} +$$$$53\!\cdots\!82$$$$p^{18} T^{45} -$$$$42\!\cdots\!29$$$$p^{20} T^{46} +$$$$25\!\cdots\!51$$$$p^{22} T^{47} +$$$$38\!\cdots\!53$$$$p^{24} T^{48} -$$$$32\!\cdots\!38$$$$p^{27} T^{49} -$$$$20\!\cdots\!74$$$$p^{28} T^{50} +$$$$12\!\cdots\!49$$$$p^{30} T^{51} -$$$$20\!\cdots\!91$$$$p^{32} T^{52} -$$$$10\!\cdots\!49$$$$p^{34} T^{53} +$$$$18\!\cdots\!02$$$$p^{36} T^{54} +$$$$58\!\cdots\!95$$$$p^{38} T^{55} -$$$$24\!\cdots\!07$$$$p^{40} T^{56} -$$$$88\!\cdots\!91$$$$p^{42} T^{57} +$$$$19\!\cdots\!96$$$$p^{44} T^{58} -$$$$22\!\cdots\!72$$$$p^{46} T^{59} -$$$$10\!\cdots\!52$$$$p^{48} T^{60} + 2997102683076161391 p^{50} T^{61} + 30972045151146096 p^{52} T^{62} - 2246797049860179 p^{54} T^{63} + 9344152756674 p^{56} T^{64} + 1124338521924 p^{58} T^{65} - 17197385286 p^{60} T^{66} - 352435905 p^{62} T^{67} + 10962684 p^{64} T^{68} + 48465 p^{66} T^{69} - 3843 p^{68} T^{70} + 18 p^{70} T^{71} + p^{72} T^{72}$$
31 $$1 - 45 T + 2961 T^{2} - 161934 T^{3} + 4602510 T^{4} - 199162188 T^{5} + 5295706530 T^{6} - 49215383559 T^{7} + 1824753317634 T^{8} + 21437058811348 T^{9} - 2573751505556430 T^{10} - 77234086448093511 T^{11} + 3803532601775738118 T^{12} -$$$$16\!\cdots\!10$$$$T^{13} +$$$$13\!\cdots\!97$$$$T^{14} -$$$$32\!\cdots\!73$$$$T^{15} +$$$$75\!\cdots\!91$$$$T^{16} -$$$$32\!\cdots\!06$$$$T^{17} +$$$$16\!\cdots\!39$$$$T^{18} +$$$$94\!\cdots\!48$$$$T^{19} +$$$$11\!\cdots\!39$$$$T^{20} +$$$$14\!\cdots\!35$$$$T^{21} +$$$$26\!\cdots\!96$$$$T^{22} -$$$$34\!\cdots\!54$$$$T^{23} +$$$$73\!\cdots\!18$$$$T^{24} -$$$$40\!\cdots\!17$$$$T^{25} +$$$$18\!\cdots\!35$$$$T^{26} -$$$$17\!\cdots\!54$$$$T^{27} +$$$$57\!\cdots\!28$$$$T^{28} -$$$$15\!\cdots\!27$$$$T^{29} -$$$$83\!\cdots\!80$$$$T^{30} +$$$$81\!\cdots\!58$$$$T^{31} +$$$$23\!\cdots\!56$$$$T^{32} +$$$$88\!\cdots\!60$$$$T^{33} +$$$$98\!\cdots\!24$$$$T^{34} -$$$$45\!\cdots\!59$$$$T^{35} +$$$$69\!\cdots\!26$$$$T^{36} -$$$$45\!\cdots\!59$$$$p^{2} T^{37} +$$$$98\!\cdots\!24$$$$p^{4} T^{38} +$$$$88\!\cdots\!60$$$$p^{6} T^{39} +$$$$23\!\cdots\!56$$$$p^{8} T^{40} +$$$$81\!\cdots\!58$$$$p^{10} T^{41} -$$$$83\!\cdots\!80$$$$p^{12} T^{42} -$$$$15\!\cdots\!27$$$$p^{14} T^{43} +$$$$57\!\cdots\!28$$$$p^{16} T^{44} -$$$$17\!\cdots\!54$$$$p^{18} T^{45} +$$$$18\!\cdots\!35$$$$p^{20} T^{46} -$$$$40\!\cdots\!17$$$$p^{22} T^{47} +$$$$73\!\cdots\!18$$$$p^{24} T^{48} -$$$$34\!\cdots\!54$$$$p^{26} T^{49} +$$$$26\!\cdots\!96$$$$p^{28} T^{50} +$$$$14\!\cdots\!35$$$$p^{30} T^{51} +$$$$11\!\cdots\!39$$$$p^{32} T^{52} +$$$$94\!\cdots\!48$$$$p^{34} T^{53} +$$$$16\!\cdots\!39$$$$p^{36} T^{54} -$$$$32\!\cdots\!06$$$$p^{38} T^{55} +$$$$75\!\cdots\!91$$$$p^{40} T^{56} -$$$$32\!\cdots\!73$$$$p^{42} T^{57} +$$$$13\!\cdots\!97$$$$p^{44} T^{58} -$$$$16\!\cdots\!10$$$$p^{46} T^{59} + 3803532601775738118 p^{48} T^{60} - 77234086448093511 p^{50} T^{61} - 2573751505556430 p^{52} T^{62} + 21437058811348 p^{54} T^{63} + 1824753317634 p^{56} T^{64} - 49215383559 p^{58} T^{65} + 5295706530 p^{60} T^{66} - 199162188 p^{62} T^{67} + 4602510 p^{64} T^{68} - 161934 p^{66} T^{69} + 2961 p^{68} T^{70} - 45 p^{70} T^{71} + p^{72} T^{72}$$
37 $$1 - 324 p T^{2} + 302430 T^{3} + 73954944 T^{4} - 3465944748 T^{5} - 266847067353 T^{6} + 20332381077756 T^{7} + 513706888750842 T^{8} - 76768032861483776 T^{9} + 103668017589897372 T^{10} +$$$$20\!\cdots\!62$$$$T^{11} -$$$$41\!\cdots\!45$$$$T^{12} -$$$$36\!\cdots\!55$$$$T^{13} +$$$$15\!\cdots\!37$$$$T^{14} +$$$$42\!\cdots\!66$$$$T^{15} -$$$$34\!\cdots\!46$$$$T^{16} -$$$$14\!\cdots\!66$$$$T^{17} +$$$$55\!\cdots\!85$$$$T^{18} -$$$$61\!\cdots\!29$$$$T^{19} -$$$$69\!\cdots\!99$$$$T^{20} +$$$$17\!\cdots\!66$$$$T^{21} +$$$$74\!\cdots\!76$$$$T^{22} -$$$$33\!\cdots\!40$$$$T^{23} -$$$$68\!\cdots\!90$$$$T^{24} +$$$$57\!\cdots\!51$$$$T^{25} +$$$$30\!\cdots\!90$$$$T^{26} -$$$$91\!\cdots\!07$$$$T^{27} +$$$$88\!\cdots\!87$$$$T^{28} +$$$$12\!\cdots\!04$$$$T^{29} -$$$$30\!\cdots\!38$$$$T^{30} -$$$$11\!\cdots\!25$$$$T^{31} +$$$$56\!\cdots\!20$$$$T^{32} +$$$$79\!\cdots\!05$$$$T^{33} -$$$$78\!\cdots\!39$$$$T^{34} -$$$$27\!\cdots\!42$$$$T^{35} +$$$$10\!\cdots\!60$$$$T^{36} -$$$$27\!\cdots\!42$$$$p^{2} T^{37} -$$$$78\!\cdots\!39$$$$p^{4} T^{38} +$$$$79\!\cdots\!05$$$$p^{6} T^{39} +$$$$56\!\cdots\!20$$$$p^{8} T^{40} -$$$$11\!\cdots\!25$$$$p^{10} T^{41} -$$$$30\!\cdots\!38$$$$p^{12} T^{42} +$$$$12\!\cdots\!04$$$$p^{14} T^{43} +$$$$88\!\cdots\!87$$$$p^{16} T^{44} -$$$$91\!\cdots\!07$$$$p^{18} T^{45} +$$$$30\!\cdots\!90$$$$p^{20} T^{46} +$$$$57\!\cdots\!51$$$$p^{22} T^{47} -$$$$68\!\cdots\!90$$$$p^{24} T^{48} -$$$$33\!\cdots\!40$$$$p^{26} T^{49} +$$$$74\!\cdots\!76$$$$p^{28} T^{50} +$$$$17\!\cdots\!66$$$$p^{30} T^{51} -$$$$69\!\cdots\!99$$$$p^{32} T^{52} -$$$$61\!\cdots\!29$$$$p^{34} T^{53} +$$$$55\!\cdots\!85$$$$p^{36} T^{54} -$$$$14\!\cdots\!66$$$$p^{38} T^{55} -$$$$34\!\cdots\!46$$$$p^{40} T^{56} +$$$$42\!\cdots\!66$$$$p^{42} T^{57} +$$$$15\!\cdots\!37$$$$p^{44} T^{58} -$$$$36\!\cdots\!55$$$$p^{46} T^{59} -$$$$41\!\cdots\!45$$$$p^{48} T^{60} +$$$$20\!\cdots\!62$$$$p^{50} T^{61} + 103668017589897372 p^{52} T^{62} - 76768032861483776 p^{54} T^{63} + 513706888750842 p^{56} T^{64} + 20332381077756 p^{58} T^{65} - 266847067353 p^{60} T^{66} - 3465944748 p^{62} T^{67} + 73954944 p^{64} T^{68} + 302430 p^{66} T^{69} - 324 p^{69} T^{70} + p^{72} T^{72}$$
41 $$1 + 198T + 1.75e4T^{2} + 5.81e5T^{3} - 3.71e7T^{4} - 5.38e9T^{5} - 2.45e11T^{6} + 1.19e12T^{7} + 7.59e14T^{8} + 3.86e16T^{9} + 2.62e17T^{10} - 6.55e19T^{11} - 3.37e21T^{12} - 3.47e22T^{13} + 3.22e24T^{14} + 1.40e26T^{15} + 1.28e27T^{16} - 2.26e27T^{17} + 4.14e30T^{18} + 1.28e32T^{19} - 1.13e34T^{20} - 9.14e35T^{21} - 1.87e37T^{22} + 6.20e38T^{23} + 4.54e40T^{24} + 1.11e42T^{25} + 1.70e43T^{26} + 4.02e43T^{27} - 5.72e46T^{28} - 4.30e48T^{29} - 7.81e49T^{30} + 4.53e51T^{31} + 2.05e53T^{32} - 3.71e54T^{33} - 3.43e56T^{34} + 4.29e57T^{35}+O(T^{36})$$
43 $$1 - 90T - 5.09e3T^{2} + 6.65e5T^{3} + 1.55e7T^{4} - 2.94e9T^{5} - 2.84e10T^{6} + 9.76e12T^{7} - 3.74e11T^{8} - 2.61e16T^{9} + 2.31e17T^{10} + 5.89e19T^{11} - 1.06e21T^{12} - 1.15e23T^{13} + 3.26e24T^{14} + 2.02e26T^{15} - 8.30e27T^{16} - 3.25e29T^{17} + 1.88e31T^{18} + 4.80e32T^{19} - 3.99e34T^{20} - 6.19e35T^{21} + 8.08e37T^{22} + 5.66e38T^{23} - 1.57e41T^{24} + 1.28e41T^{25} + 2.89e44T^{26} - 2.26e45T^{27} - 4.92e47T^{28} + 6.59e48T^{29} + 7.77e50T^{30} - 1.29e52T^{31} - 1.15e54T^{32} + 1.81e55T^{33} + 1.72e57T^{34} - 1.24e58T^{35}+O(T^{36})$$
47 $$1 + 243T + 3.71e4T^{2} + 4.42e6T^{3} + 4.21e8T^{4} + 3.40e10T^{5} + 2.38e12T^{6} + 1.44e14T^{7} + 7.92e15T^{8} + 3.94e17T^{9} + 1.87e19T^{10} + 9.02e20T^{11} + 4.55e22T^{12} + 2.40e24T^{13} + 1.25e26T^{14} + 6.07e27T^{15} + 2.68e29T^{16} + 1.07e31T^{17} + 4.20e32T^{18} + 1.82e34T^{19} + 9.22e35T^{20} + 5.00e37T^{21} + 2.56e39T^{22} + 1.14e41T^{23} + 4.44e42T^{24} + 1.60e44T^{25} + 6.18e45T^{26} + 2.95e47T^{27} + 1.55e49T^{28} + 7.14e50T^{29} + 2.49e52T^{30} + 4.79e53T^{31} - 9.71e54T^{32} - 1.08e57T^{33} - 4.11e58T^{34}+O(T^{35})$$
53 $$1 - 4.62e4T^{2} + 1.11e9T^{4} - 1.83e13T^{6} + 2.32e17T^{8} - 2.41e21T^{10} + 2.11e25T^{12} - 1.61e29T^{14} + 1.08e33T^{16} - 6.53e36T^{18} + 3.56e40T^{20} - 1.77e44T^{22} + 8.08e47T^{24} - 3.39e51T^{26} + 1.32e55T^{28} - 4.76e58T^{30} + 1.59e62T^{32}+O(T^{34})$$
59 $$1 - 252T + 2.05e4T^{2} - 4.39e5T^{3} + 2.08e7T^{4} - 4.42e9T^{5} - 3.01e10T^{6} + 2.17e13T^{7} + 4.93e14T^{8} - 5.06e16T^{9} - 6.29e18T^{10} + 1.60e20T^{11} + 9.75e21T^{12} + 2.15e24T^{13} - 1.21e26T^{14} - 3.81e27T^{15} - 2.89e29T^{16} + 3.62e31T^{17} + 1.41e33T^{18} - 4.84e34T^{19} - 7.11e36T^{20} - 3.25e38T^{21} + 3.69e40T^{22} + 6.96e41T^{23} + 4.39e43T^{24} - 9.17e45T^{25} - 1.08e47T^{26} + 8.34e47T^{27} + 2.05e51T^{28} + 9.70e51T^{29} - 3.72e54T^{30} - 2.75e56T^{31} - 3.29e57T^{32}+O(T^{33})$$
61 $$1 + 144T + 1.04e4T^{2} + 6.91e5T^{3} + 8.44e7T^{4} + 8.29e9T^{5} + 5.74e11T^{6} + 4.19e13T^{7} + 3.52e15T^{8} + 2.88e17T^{9} + 2.09e19T^{10} + 1.45e21T^{11} + 9.93e22T^{12} + 7.25e24T^{13} + 5.14e26T^{14} + 3.17e28T^{15} + 2.00e30T^{16} + 1.35e32T^{17} + 8.64e33T^{18} + 4.94e35T^{19} + 2.92e37T^{20} + 1.70e39T^{21} + 9.32e40T^{22} + 4.95e42T^{23} + 2.26e44T^{24} + 8.91e45T^{25} + 3.11e47T^{26} - 1.61e47T^{27} - 1.62e51T^{28} - 1.58e53T^{29} - 1.29e55T^{30} - 1.12e57T^{31} - 8.15e58T^{32}+O(T^{33})$$
67 $$1 - 108T - 3.78e3T^{2} + 4.84e5T^{3} - 6.95e7T^{4} + 9.19e9T^{5} + 2.82e11T^{6} - 4.12e13T^{7} + 4.85e14T^{8} - 3.27e17T^{9} + 1.87e18T^{10} + 1.35e21T^{11} + 7.83e22T^{12} + 5.48e24T^{13} - 7.52e26T^{14} - 1.66e28T^{15} - 2.62e30T^{16} + 3.96e31T^{17} + 2.77e34T^{18} - 2.75e35T^{19} + 2.47e37T^{20} - 5.15e39T^{21} - 3.30e41T^{22} + 1.57e43T^{23} + 3.15e44T^{24} + 1.32e47T^{25} - 5.78e48T^{26} - 2.03e50T^{27} - 9.10e51T^{28} - 1.01e54T^{29} + 2.66e56T^{30} - 4.55e57T^{31}+O(T^{32})$$
71 $$1 - 324T + 9.11e4T^{2} - 1.82e7T^{3} + 3.17e9T^{4} - 4.77e11T^{5} + 6.48e13T^{6} - 8.07e15T^{7} + 9.36e17T^{8} - 1.02e20T^{9} + 1.07e22T^{10} - 1.07e24T^{11} + 1.04e26T^{12} - 9.92e27T^{13} + 9.15e29T^{14} - 8.26e31T^{15} + 7.31e33T^{16} - 6.35e35T^{17} + 5.43e37T^{18} - 4.58e39T^{19} + 3.81e41T^{20} - 3.13e43T^{21} + 2.54e45T^{22} - 2.04e47T^{23} + 1.62e49T^{24} - 1.27e51T^{25} + 9.90e52T^{26} - 7.62e54T^{27} + 5.82e56T^{28} - 4.40e58T^{29} + 3.29e60T^{30} - 2.45e62T^{31}+O(T^{32})$$
73 $$1 + 63T - 4.05e4T^{2} - 1.28e6T^{3} + 9.37e8T^{4} + 4.01e9T^{5} - 1.47e13T^{6} + 2.97e14T^{7} + 1.71e17T^{8} - 7.57e18T^{9} - 1.50e21T^{10} + 1.08e23T^{11} + 9.79e24T^{12} - 1.11e27T^{13} - 4.13e28T^{14} + 8.87e30T^{15} + 2.52e31T^{16} - 5.54e34T^{17} + 1.33e36T^{18} + 2.71e38T^{19} - 1.34e40T^{20} - 1.01e42T^{21} + 8.00e43T^{22} + 3.04e45T^{23} - 3.25e47T^{24} - 1.22e49T^{25} + 9.81e50T^{26} + 1.08e53T^{27} - 4.31e54T^{28} - 9.85e56T^{29} + 4.81e58T^{30}+O(T^{31})$$
79 $$1 - 36T + 9.42e3T^{2} - 5.36e5T^{3} + 6.16e6T^{4} - 1.76e9T^{5} - 6.92e10T^{6} + 8.62e12T^{7} + 1.26e15T^{8} + 1.44e17T^{9} + 4.55e17T^{10} + 1.02e21T^{11} - 1.45e23T^{12} + 4.61e24T^{13} - 8.31e26T^{14} + 3.34e28T^{15} + 1.15e30T^{16} + 3.07e32T^{17} + 9.09e33T^{18} + 2.38e35T^{19} - 1.16e38T^{20} - 1.30e40T^{21} + 2.39e41T^{22} - 8.94e43T^{23} + 1.27e46T^{24} - 4.19e47T^{25} + 6.53e49T^{26} - 2.43e51T^{27} - 2.99e52T^{28} - 1.06e55T^{29} - 7.00e56T^{30}+O(T^{31})$$
83 $$1 + 27T + 1.60e4T^{2} + 1.65e6T^{3} + 2.68e8T^{4} + 2.72e10T^{5} + 4.10e12T^{6} + 3.81e14T^{7} + 5.20e16T^{8} + 4.84e18T^{9} + 5.66e20T^{10} + 5.43e22T^{11} + 5.93e24T^{12} + 5.28e26T^{13} + 5.76e28T^{14} + 4.95e30T^{15} + 5.08e32T^{16} + 4.40e34T^{17} + 4.35e36T^{18} + 3.64e38T^{19} + 3.62e40T^{20} + 2.94e42T^{21} + 2.87e44T^{22} + 2.35e46T^{23} + 2.24e48T^{24} + 1.83e50T^{25} + 1.74e52T^{26} + 1.39e54T^{27} + 1.32e56T^{28} + 1.07e58T^{29} + 9.92e59T^{30}+O(T^{31})$$
89 $$1 + 567T + 2.17e5T^{2} + 6.25e7T^{3} + 1.49e10T^{4} + 3.07e12T^{5} + 5.63e14T^{6} + 9.36e16T^{7} + 1.43e19T^{8} + 2.06e21T^{9} + 2.81e23T^{10} + 3.67e25T^{11} + 4.65e27T^{12} + 5.74e29T^{13} + 6.92e31T^{14} + 8.17e33T^{15} + 9.46e35T^{16} + 1.07e38T^{17} + 1.19e40T^{18} + 1.31e42T^{19} + 1.41e44T^{20} + 1.50e46T^{21} + 1.57e48T^{22} + 1.62e50T^{23} + 1.66e52T^{24} + 1.67e54T^{25} + 1.67e56T^{26} + 1.64e58T^{27} + 1.59e60T^{28} + 1.53e62T^{29}+O(T^{30})$$
97 $$1 + 216T + 3.39e4T^{2} + 3.96e6T^{3} + 1.51e8T^{4} - 9.22e9T^{5} - 3.75e12T^{6} - 3.80e14T^{7} + 2.38e15T^{8} + 3.50e18T^{9} + 7.27e20T^{10} + 5.66e22T^{11} + 1.09e24T^{12} - 1.79e26T^{13} - 4.36e28T^{14} - 3.70e29T^{15} + 3.18e32T^{16} + 4.93e34T^{17} + 5.25e36T^{18} + 2.56e37T^{19} - 9.49e39T^{20} - 1.80e42T^{21} - 1.20e44T^{22} + 1.12e46T^{23} - 9.93e47T^{24} - 1.15e50T^{25} - 1.50e52T^{26} + 1.15e53T^{27} + 4.09e56T^{28} + 4.29e58T^{29}+O(T^{30})$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{72} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}