# Properties

 Degree 160 Conductor $2^{80} \cdot 3^{80} \cdot 5^{160}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4·3-s + 4·7-s + 8·9-s + 5·16-s − 40·19-s + 16·21-s − 52·25-s + 12·27-s − 24·37-s − 24·43-s + 20·48-s + 8·49-s − 160·57-s + 32·63-s + 96·67-s + 100·73-s − 208·75-s + 80·79-s + 6·81-s − 32·97-s − 124·103-s − 40·109-s − 96·111-s + 20·112-s − 100·121-s + 127-s − 96·129-s + ⋯
 L(s)  = 1 + 2.30·3-s + 1.51·7-s + 8/3·9-s + 5/4·16-s − 9.17·19-s + 3.49·21-s − 10.3·25-s + 2.30·27-s − 3.94·37-s − 3.65·43-s + 2.88·48-s + 8/7·49-s − 21.1·57-s + 4.03·63-s + 11.7·67-s + 11.7·73-s − 24.0·75-s + 9.00·79-s + 2/3·81-s − 3.24·97-s − 12.2·103-s − 3.83·109-s − 9.11·111-s + 1.88·112-s − 9.09·121-s + 0.0887·127-s − 8.45·129-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{80} \cdot 3^{80} \cdot 5^{160}\right)^{s/2} \, \Gamma_{\C}(s)^{80} \, L(s)\cr =\mathstrut & \,\Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{80} \cdot 3^{80} \cdot 5^{160}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{80} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$160$$ $$N$$ = $$2^{80} \cdot 3^{80} \cdot 5^{160}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : induced by $\chi_{150} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(160,\ 2^{80} \cdot 3^{80} \cdot 5^{160} ,\ ( \ : [1/2]^{80} ),\ 1 )$ $L(1)$ $\approx$ $0.0377704$ $L(\frac12)$ $\approx$ $0.0377704$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3,\;5\}$, $$F_p$$ is a polynomial of degree 160. If $p \in \{2,\;3,\;5\}$, then $F_p$ is a polynomial of degree at most 159.
$p$$F_p$
bad2 $$( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{5}$$
3 $$1 - 4 T + 8 T^{2} - 4 p T^{3} + 26 T^{4} - 104 T^{5} + 290 T^{6} - 572 T^{7} + 998 T^{8} - 1888 T^{9} + 500 p^{2} T^{10} - 10184 T^{11} + 20482 T^{12} - 38180 T^{13} + 65186 T^{14} - 13720 p^{2} T^{15} + 252590 T^{16} - 513364 T^{17} + 106280 p^{2} T^{18} - 1590532 T^{19} + 2839744 T^{20} - 1810460 p T^{21} + 10608968 T^{22} - 19815700 T^{23} + 34323986 T^{24} - 20422888 p T^{25} + 110536294 T^{26} - 206568772 T^{27} + 386537870 T^{28} - 8434360 p^{4} T^{29} + 5005940 p^{5} T^{30} - 715522960 p T^{31} + 3919220570 T^{32} - 7085639516 T^{33} + 12363034406 T^{34} - 22061371688 T^{35} + 12853452242 p T^{36} - 22756753220 p T^{37} + 119341180072 T^{38} - 68812467100 p T^{39} + 363108840430 T^{40} - 68812467100 p^{2} T^{41} + 119341180072 p^{2} T^{42} - 22756753220 p^{4} T^{43} + 12853452242 p^{5} T^{44} - 22061371688 p^{5} T^{45} + 12363034406 p^{6} T^{46} - 7085639516 p^{7} T^{47} + 3919220570 p^{8} T^{48} - 715522960 p^{10} T^{49} + 5005940 p^{15} T^{50} - 8434360 p^{15} T^{51} + 386537870 p^{12} T^{52} - 206568772 p^{13} T^{53} + 110536294 p^{14} T^{54} - 20422888 p^{16} T^{55} + 34323986 p^{16} T^{56} - 19815700 p^{17} T^{57} + 10608968 p^{18} T^{58} - 1810460 p^{20} T^{59} + 2839744 p^{20} T^{60} - 1590532 p^{21} T^{61} + 106280 p^{24} T^{62} - 513364 p^{23} T^{63} + 252590 p^{24} T^{64} - 13720 p^{27} T^{65} + 65186 p^{26} T^{66} - 38180 p^{27} T^{67} + 20482 p^{28} T^{68} - 10184 p^{29} T^{69} + 500 p^{32} T^{70} - 1888 p^{31} T^{71} + 998 p^{32} T^{72} - 572 p^{33} T^{73} + 290 p^{34} T^{74} - 104 p^{35} T^{75} + 26 p^{36} T^{76} - 4 p^{38} T^{77} + 8 p^{38} T^{78} - 4 p^{39} T^{79} + p^{40} T^{80}$$
5 $$1 + 52 T^{2} + 1399 T^{4} + 25888 T^{6} + 367851 T^{8} + 169608 p^{2} T^{10} + 8169641 p T^{12} + 66739692 p T^{14} + 463084782 p T^{16} + 107488364 p^{3} T^{18} + 2475460651 p^{2} T^{20} + 7469391696 p^{2} T^{22} - 4024369199 p^{2} T^{24} - 51036624368 p^{3} T^{26} - 17242312731 p^{5} T^{28} - 92397169452 p^{5} T^{30} - 304881931868 p^{5} T^{32} + 2380478492 p^{7} T^{34} + 389588104459 p^{7} T^{36} + 138777499704 p^{9} T^{38} + 4051344989581 p^{8} T^{40} + 138777499704 p^{11} T^{42} + 389588104459 p^{11} T^{44} + 2380478492 p^{13} T^{46} - 304881931868 p^{13} T^{48} - 92397169452 p^{15} T^{50} - 17242312731 p^{17} T^{52} - 51036624368 p^{17} T^{54} - 4024369199 p^{18} T^{56} + 7469391696 p^{20} T^{58} + 2475460651 p^{22} T^{60} + 107488364 p^{25} T^{62} + 463084782 p^{25} T^{64} + 66739692 p^{27} T^{66} + 8169641 p^{29} T^{68} + 169608 p^{32} T^{70} + 367851 p^{32} T^{72} + 25888 p^{34} T^{74} + 1399 p^{36} T^{76} + 52 p^{38} T^{78} + p^{40} T^{80}$$
good7 $$( 1 - 2 T + 2 T^{2} + 6 T^{3} - p^{2} T^{4} - 262 T^{5} + 640 T^{6} - 2524 T^{7} - 251 p T^{8} + 14836 T^{9} + 24380 T^{10} - 107138 T^{11} + 892057 T^{12} + 527890 T^{13} - 1865366 T^{14} + 8892670 T^{15} + 3641660 T^{16} - 153084902 T^{17} - 25078670 T^{18} - 146774714 T^{19} - 3342408911 T^{20} - 35668730 p T^{21} + 8952413612 T^{22} - 18830878100 T^{23} + 54439958341 T^{24} + 582647890168 T^{25} + 73114178756 T^{26} + 36435910874 p^{2} T^{27} + 7528475287545 T^{28} - 4534697526410 T^{29} - 34528354057670 T^{30} + 7105751150 p T^{31} - 425856152031425 T^{32} - 171310619263144 p T^{33} - 964109762455496 T^{34} - 5023921487039956 T^{35} - 8496593660556514 T^{36} + 28482794632191140 T^{37} + 1540739250407972 p^{2} T^{38} + 31227987396083300 p T^{39} + 1374314483413633000 T^{40} + 31227987396083300 p^{2} T^{41} + 1540739250407972 p^{4} T^{42} + 28482794632191140 p^{3} T^{43} - 8496593660556514 p^{4} T^{44} - 5023921487039956 p^{5} T^{45} - 964109762455496 p^{6} T^{46} - 171310619263144 p^{8} T^{47} - 425856152031425 p^{8} T^{48} + 7105751150 p^{10} T^{49} - 34528354057670 p^{10} T^{50} - 4534697526410 p^{11} T^{51} + 7528475287545 p^{12} T^{52} + 36435910874 p^{15} T^{53} + 73114178756 p^{14} T^{54} + 582647890168 p^{15} T^{55} + 54439958341 p^{16} T^{56} - 18830878100 p^{17} T^{57} + 8952413612 p^{18} T^{58} - 35668730 p^{20} T^{59} - 3342408911 p^{20} T^{60} - 146774714 p^{21} T^{61} - 25078670 p^{22} T^{62} - 153084902 p^{23} T^{63} + 3641660 p^{24} T^{64} + 8892670 p^{25} T^{65} - 1865366 p^{26} T^{66} + 527890 p^{27} T^{67} + 892057 p^{28} T^{68} - 107138 p^{29} T^{69} + 24380 p^{30} T^{70} + 14836 p^{31} T^{71} - 251 p^{33} T^{72} - 2524 p^{33} T^{73} + 640 p^{34} T^{74} - 262 p^{35} T^{75} - p^{38} T^{76} + 6 p^{37} T^{77} + 2 p^{38} T^{78} - 2 p^{39} T^{79} + p^{40} T^{80} )^{2}$$
11 $$1 + 100 T^{2} + 4445 T^{4} + 106100 T^{6} + 1046720 T^{8} - 15752400 T^{10} - 721142830 T^{12} - 10307556900 T^{14} - 8830586330 T^{16} + 2217385146600 T^{18} + 37965801159883 T^{20} + 160120365846700 T^{22} - 4315079749732110 T^{24} - 84366642864726800 T^{26} - 423716470699628785 T^{28} + 635685277744691200 p T^{30} +$$$$13\!\cdots\!90$$$$T^{32} +$$$$54\!\cdots\!00$$$$T^{34} -$$$$12\!\cdots\!10$$$$T^{36} -$$$$19\!\cdots\!00$$$$T^{38} -$$$$32\!\cdots\!97$$$$T^{40} +$$$$23\!\cdots\!00$$$$T^{42} +$$$$28\!\cdots\!20$$$$T^{44} -$$$$45\!\cdots\!00$$$$T^{46} -$$$$35\!\cdots\!30$$$$T^{48} -$$$$37\!\cdots\!00$$$$T^{50} +$$$$11\!\cdots\!95$$$$T^{52} +$$$$41\!\cdots\!00$$$$T^{54} +$$$$33\!\cdots\!70$$$$p T^{56} -$$$$60\!\cdots\!00$$$$T^{58} -$$$$37\!\cdots\!19$$$$T^{60} -$$$$20\!\cdots\!00$$$$T^{62} +$$$$23\!\cdots\!50$$$$T^{64} +$$$$35\!\cdots\!00$$$$T^{66} -$$$$11\!\cdots\!00$$$$T^{68} -$$$$43\!\cdots\!00$$$$T^{70} -$$$$37\!\cdots\!00$$$$T^{72} +$$$$68\!\cdots\!00$$$$T^{74} +$$$$33\!\cdots\!25$$$$T^{76} +$$$$15\!\cdots\!00$$$$T^{78} -$$$$36\!\cdots\!80$$$$T^{80} +$$$$15\!\cdots\!00$$$$p^{2} T^{82} +$$$$33\!\cdots\!25$$$$p^{4} T^{84} +$$$$68\!\cdots\!00$$$$p^{6} T^{86} -$$$$37\!\cdots\!00$$$$p^{8} T^{88} -$$$$43\!\cdots\!00$$$$p^{10} T^{90} -$$$$11\!\cdots\!00$$$$p^{12} T^{92} +$$$$35\!\cdots\!00$$$$p^{14} T^{94} +$$$$23\!\cdots\!50$$$$p^{16} T^{96} -$$$$20\!\cdots\!00$$$$p^{18} T^{98} -$$$$37\!\cdots\!19$$$$p^{20} T^{100} -$$$$60\!\cdots\!00$$$$p^{22} T^{102} +$$$$33\!\cdots\!70$$$$p^{25} T^{104} +$$$$41\!\cdots\!00$$$$p^{26} T^{106} +$$$$11\!\cdots\!95$$$$p^{28} T^{108} -$$$$37\!\cdots\!00$$$$p^{30} T^{110} -$$$$35\!\cdots\!30$$$$p^{32} T^{112} -$$$$45\!\cdots\!00$$$$p^{34} T^{114} +$$$$28\!\cdots\!20$$$$p^{36} T^{116} +$$$$23\!\cdots\!00$$$$p^{38} T^{118} -$$$$32\!\cdots\!97$$$$p^{40} T^{120} -$$$$19\!\cdots\!00$$$$p^{42} T^{122} -$$$$12\!\cdots\!10$$$$p^{44} T^{124} +$$$$54\!\cdots\!00$$$$p^{46} T^{126} +$$$$13\!\cdots\!90$$$$p^{48} T^{128} + 635685277744691200 p^{51} T^{130} - 423716470699628785 p^{52} T^{132} - 84366642864726800 p^{54} T^{134} - 4315079749732110 p^{56} T^{136} + 160120365846700 p^{58} T^{138} + 37965801159883 p^{60} T^{140} + 2217385146600 p^{62} T^{142} - 8830586330 p^{64} T^{144} - 10307556900 p^{66} T^{146} - 721142830 p^{68} T^{148} - 15752400 p^{70} T^{150} + 1046720 p^{72} T^{152} + 106100 p^{74} T^{154} + 4445 p^{76} T^{156} + 100 p^{78} T^{158} + p^{80} T^{160}$$
13 $$( 1 + 20 T^{2} + 80 T^{3} + 240 T^{4} + 860 T^{5} + 20 p^{2} T^{6} - 13860 T^{7} - 6295 T^{8} - 544800 T^{9} - 1725700 T^{10} - 7149000 T^{11} - 25482455 T^{12} - 48604420 T^{13} + 129412430 T^{14} + 626885580 T^{15} + 9804607885 T^{16} + 29940705060 T^{17} + 137826752920 T^{18} + 444201423600 T^{19} + 832836734021 T^{20} - 109602655700 T^{21} - 10136340605310 T^{22} - 99313248775840 T^{23} - 329180876007370 T^{24} - 1659553730690280 T^{25} - 4263148516660140 T^{26} - 9799737378988920 T^{27} - 6417014996298215 T^{28} + 96710757403344100 T^{29} + 761316115959989350 T^{30} + 2893978720902799900 T^{31} + 15503062750157718390 T^{32} + 35510858770169506760 T^{33} +$$$$12\!\cdots\!10$$$$T^{34} +$$$$19\!\cdots\!60$$$$T^{35} - 37816797591463587930 T^{36} -$$$$30\!\cdots\!80$$$$T^{37} -$$$$16\!\cdots\!10$$$$T^{38} -$$$$10\!\cdots\!00$$$$T^{39} -$$$$25\!\cdots\!19$$$$T^{40} -$$$$10\!\cdots\!00$$$$p T^{41} -$$$$16\!\cdots\!10$$$$p^{2} T^{42} -$$$$30\!\cdots\!80$$$$p^{3} T^{43} - 37816797591463587930 p^{4} T^{44} +$$$$19\!\cdots\!60$$$$p^{5} T^{45} +$$$$12\!\cdots\!10$$$$p^{6} T^{46} + 35510858770169506760 p^{7} T^{47} + 15503062750157718390 p^{8} T^{48} + 2893978720902799900 p^{9} T^{49} + 761316115959989350 p^{10} T^{50} + 96710757403344100 p^{11} T^{51} - 6417014996298215 p^{12} T^{52} - 9799737378988920 p^{13} T^{53} - 4263148516660140 p^{14} T^{54} - 1659553730690280 p^{15} T^{55} - 329180876007370 p^{16} T^{56} - 99313248775840 p^{17} T^{57} - 10136340605310 p^{18} T^{58} - 109602655700 p^{19} T^{59} + 832836734021 p^{20} T^{60} + 444201423600 p^{21} T^{61} + 137826752920 p^{22} T^{62} + 29940705060 p^{23} T^{63} + 9804607885 p^{24} T^{64} + 626885580 p^{25} T^{65} + 129412430 p^{26} T^{66} - 48604420 p^{27} T^{67} - 25482455 p^{28} T^{68} - 7149000 p^{29} T^{69} - 1725700 p^{30} T^{70} - 544800 p^{31} T^{71} - 6295 p^{32} T^{72} - 13860 p^{33} T^{73} + 20 p^{36} T^{74} + 860 p^{35} T^{75} + 240 p^{36} T^{76} + 80 p^{37} T^{77} + 20 p^{38} T^{78} + p^{40} T^{80} )^{2}$$
17 $$1 - 40 T^{2} + 2280 T^{4} - 66800 T^{6} + 2268320 T^{8} - 53660800 T^{10} + 1341688760 T^{12} - 27707248600 T^{14} + 577474403760 T^{16} - 12080496982560 T^{18} + 246398821889688 T^{20} - 5825593667650480 T^{22} + 123843511080068640 T^{24} - 2920929045228226200 T^{26} + 59494034593832395480 T^{28} -$$$$12\!\cdots\!00$$$$T^{30} +$$$$14\!\cdots\!40$$$$p T^{32} -$$$$45\!\cdots\!00$$$$T^{34} +$$$$87\!\cdots\!40$$$$T^{36} -$$$$16\!\cdots\!20$$$$T^{38} +$$$$34\!\cdots\!08$$$$T^{40} -$$$$66\!\cdots\!40$$$$T^{42} +$$$$13\!\cdots\!60$$$$T^{44} -$$$$25\!\cdots\!00$$$$T^{46} +$$$$48\!\cdots\!60$$$$T^{48} -$$$$87\!\cdots\!00$$$$T^{50} +$$$$15\!\cdots\!20$$$$T^{52} -$$$$28\!\cdots\!00$$$$T^{54} +$$$$52\!\cdots\!80$$$$T^{56} -$$$$98\!\cdots\!60$$$$T^{58} +$$$$18\!\cdots\!56$$$$T^{60} -$$$$33\!\cdots\!00$$$$T^{62} +$$$$60\!\cdots\!00$$$$T^{64} -$$$$10\!\cdots\!00$$$$T^{66} +$$$$18\!\cdots\!00$$$$T^{68} -$$$$30\!\cdots\!00$$$$T^{70} +$$$$53\!\cdots\!00$$$$T^{72} -$$$$94\!\cdots\!00$$$$T^{74} +$$$$16\!\cdots\!00$$$$T^{76} -$$$$29\!\cdots\!00$$$$T^{78} +$$$$50\!\cdots\!70$$$$T^{80} -$$$$29\!\cdots\!00$$$$p^{2} T^{82} +$$$$16\!\cdots\!00$$$$p^{4} T^{84} -$$$$94\!\cdots\!00$$$$p^{6} T^{86} +$$$$53\!\cdots\!00$$$$p^{8} T^{88} -$$$$30\!\cdots\!00$$$$p^{10} T^{90} +$$$$18\!\cdots\!00$$$$p^{12} T^{92} -$$$$10\!\cdots\!00$$$$p^{14} T^{94} +$$$$60\!\cdots\!00$$$$p^{16} T^{96} -$$$$33\!\cdots\!00$$$$p^{18} T^{98} +$$$$18\!\cdots\!56$$$$p^{20} T^{100} -$$$$98\!\cdots\!60$$$$p^{22} T^{102} +$$$$52\!\cdots\!80$$$$p^{24} T^{104} -$$$$28\!\cdots\!00$$$$p^{26} T^{106} +$$$$15\!\cdots\!20$$$$p^{28} T^{108} -$$$$87\!\cdots\!00$$$$p^{30} T^{110} +$$$$48\!\cdots\!60$$$$p^{32} T^{112} -$$$$25\!\cdots\!00$$$$p^{34} T^{114} +$$$$13\!\cdots\!60$$$$p^{36} T^{116} -$$$$66\!\cdots\!40$$$$p^{38} T^{118} +$$$$34\!\cdots\!08$$$$p^{40} T^{120} -$$$$16\!\cdots\!20$$$$p^{42} T^{122} +$$$$87\!\cdots\!40$$$$p^{44} T^{124} -$$$$45\!\cdots\!00$$$$p^{46} T^{126} +$$$$14\!\cdots\!40$$$$p^{49} T^{128} -$$$$12\!\cdots\!00$$$$p^{50} T^{130} + 59494034593832395480 p^{52} T^{132} - 2920929045228226200 p^{54} T^{134} + 123843511080068640 p^{56} T^{136} - 5825593667650480 p^{58} T^{138} + 246398821889688 p^{60} T^{140} - 12080496982560 p^{62} T^{142} + 577474403760 p^{64} T^{144} - 27707248600 p^{66} T^{146} + 1341688760 p^{68} T^{148} - 53660800 p^{70} T^{150} + 2268320 p^{72} T^{152} - 66800 p^{74} T^{154} + 2280 p^{76} T^{156} - 40 p^{78} T^{158} + p^{80} T^{160}$$
19 $$( 1 + 20 T + 260 T^{2} + 2440 T^{3} + 18735 T^{4} + 121100 T^{5} + 695410 T^{6} + 3616900 T^{7} + 17665335 T^{8} + 82315540 T^{9} + 373086522 T^{10} + 1650214200 T^{11} + 7122042375 T^{12} + 29813150400 T^{13} + 118918557680 T^{14} + 445103443800 T^{15} + 1465145238380 T^{16} + 3736406321100 T^{17} + 1780802417960 T^{18} - 61867867274200 T^{19} - 593141414596067 T^{20} - 3916302771803760 T^{21} - 22465676151010650 T^{22} - 117645537856495700 T^{23} - 582162418354057525 T^{24} - 2738839918301717800 T^{25} - 12322201048562974650 T^{26} - 52979243970628299160 T^{27} -$$$$21\!\cdots\!25$$$$T^{28} -$$$$83\!\cdots\!80$$$$T^{29} -$$$$29\!\cdots\!96$$$$T^{30} -$$$$85\!\cdots\!00$$$$T^{31} -$$$$15\!\cdots\!40$$$$T^{32} + 98812487198304756100 p^{2} T^{33} +$$$$59\!\cdots\!60$$$$T^{34} +$$$$42\!\cdots\!00$$$$T^{35} +$$$$24\!\cdots\!35$$$$T^{36} +$$$$12\!\cdots\!00$$$$T^{37} +$$$$32\!\cdots\!90$$$$p T^{38} +$$$$28\!\cdots\!00$$$$T^{39} +$$$$12\!\cdots\!05$$$$T^{40} +$$$$28\!\cdots\!00$$$$p T^{41} +$$$$32\!\cdots\!90$$$$p^{3} T^{42} +$$$$12\!\cdots\!00$$$$p^{3} T^{43} +$$$$24\!\cdots\!35$$$$p^{4} T^{44} +$$$$42\!\cdots\!00$$$$p^{5} T^{45} +$$$$59\!\cdots\!60$$$$p^{6} T^{46} + 98812487198304756100 p^{9} T^{47} -$$$$15\!\cdots\!40$$$$p^{8} T^{48} -$$$$85\!\cdots\!00$$$$p^{9} T^{49} -$$$$29\!\cdots\!96$$$$p^{10} T^{50} -$$$$83\!\cdots\!80$$$$p^{11} T^{51} -$$$$21\!\cdots\!25$$$$p^{12} T^{52} - 52979243970628299160 p^{13} T^{53} - 12322201048562974650 p^{14} T^{54} - 2738839918301717800 p^{15} T^{55} - 582162418354057525 p^{16} T^{56} - 117645537856495700 p^{17} T^{57} - 22465676151010650 p^{18} T^{58} - 3916302771803760 p^{19} T^{59} - 593141414596067 p^{20} T^{60} - 61867867274200 p^{21} T^{61} + 1780802417960 p^{22} T^{62} + 3736406321100 p^{23} T^{63} + 1465145238380 p^{24} T^{64} + 445103443800 p^{25} T^{65} + 118918557680 p^{26} T^{66} + 29813150400 p^{27} T^{67} + 7122042375 p^{28} T^{68} + 1650214200 p^{29} T^{69} + 373086522 p^{30} T^{70} + 82315540 p^{31} T^{71} + 17665335 p^{32} T^{72} + 3616900 p^{33} T^{73} + 695410 p^{34} T^{74} + 121100 p^{35} T^{75} + 18735 p^{36} T^{76} + 2440 p^{37} T^{77} + 260 p^{38} T^{78} + 20 p^{39} T^{79} + p^{40} T^{80} )^{2}$$
23 $$1 - 40 T^{2} + 200 T^{4} + 56700 T^{6} - 1299490 T^{8} - 18120400 T^{10} + 1975748970 T^{12} - 23973045400 T^{14} - 784841485165 T^{16} + 1748399615180 p T^{18} - 150891299050092 T^{20} - 20208418858575980 T^{22} + 600994833233387135 T^{24} + 2128668064666234400 T^{26} -$$$$31\!\cdots\!70$$$$T^{28} +$$$$58\!\cdots\!00$$$$T^{30} +$$$$74\!\cdots\!55$$$$T^{32} -$$$$33\!\cdots\!00$$$$T^{34} +$$$$44\!\cdots\!60$$$$T^{36} +$$$$73\!\cdots\!80$$$$T^{38} -$$$$22\!\cdots\!47$$$$T^{40} +$$$$12\!\cdots\!20$$$$p T^{42} +$$$$30\!\cdots\!10$$$$T^{44} -$$$$12\!\cdots\!00$$$$T^{46} +$$$$48\!\cdots\!20$$$$T^{48} -$$$$46\!\cdots\!00$$$$T^{50} -$$$$11\!\cdots\!40$$$$T^{52} +$$$$62\!\cdots\!00$$$$T^{54} +$$$$61\!\cdots\!75$$$$T^{56} -$$$$29\!\cdots\!60$$$$T^{58} +$$$$57\!\cdots\!06$$$$T^{60} +$$$$12\!\cdots\!00$$$$T^{62} -$$$$27\!\cdots\!00$$$$T^{64} -$$$$83\!\cdots\!00$$$$T^{66} +$$$$30\!\cdots\!00$$$$T^{68} -$$$$23\!\cdots\!00$$$$T^{70} -$$$$37\!\cdots\!00$$$$T^{72} +$$$$34\!\cdots\!00$$$$T^{74} -$$$$30\!\cdots\!50$$$$T^{76} -$$$$55\!\cdots\!00$$$$T^{78} +$$$$44\!\cdots\!45$$$$T^{80} -$$$$55\!\cdots\!00$$$$p^{2} T^{82} -$$$$30\!\cdots\!50$$$$p^{4} T^{84} +$$$$34\!\cdots\!00$$$$p^{6} T^{86} -$$$$37\!\cdots\!00$$$$p^{8} T^{88} -$$$$23\!\cdots\!00$$$$p^{10} T^{90} +$$$$30\!\cdots\!00$$$$p^{12} T^{92} -$$$$83\!\cdots\!00$$$$p^{14} T^{94} -$$$$27\!\cdots\!00$$$$p^{16} T^{96} +$$$$12\!\cdots\!00$$$$p^{18} T^{98} +$$$$57\!\cdots\!06$$$$p^{20} T^{100} -$$$$29\!\cdots\!60$$$$p^{22} T^{102} +$$$$61\!\cdots\!75$$$$p^{24} T^{104} +$$$$62\!\cdots\!00$$$$p^{26} T^{106} -$$$$11\!\cdots\!40$$$$p^{28} T^{108} -$$$$46\!\cdots\!00$$$$p^{30} T^{110} +$$$$48\!\cdots\!20$$$$p^{32} T^{112} -$$$$12\!\cdots\!00$$$$p^{34} T^{114} +$$$$30\!\cdots\!10$$$$p^{36} T^{116} +$$$$12\!\cdots\!20$$$$p^{39} T^{118} -$$$$22\!\cdots\!47$$$$p^{40} T^{120} +$$$$73\!\cdots\!80$$$$p^{42} T^{122} +$$$$44\!\cdots\!60$$$$p^{44} T^{124} -$$$$33\!\cdots\!00$$$$p^{46} T^{126} +$$$$74\!\cdots\!55$$$$p^{48} T^{128} +$$$$58\!\cdots\!00$$$$p^{50} T^{130} -$$$$31\!\cdots\!70$$$$p^{52} T^{132} + 2128668064666234400 p^{54} T^{134} + 600994833233387135 p^{56} T^{136} - 20208418858575980 p^{58} T^{138} - 150891299050092 p^{60} T^{140} + 1748399615180 p^{63} T^{142} - 784841485165 p^{64} T^{144} - 23973045400 p^{66} T^{146} + 1975748970 p^{68} T^{148} - 18120400 p^{70} T^{150} - 1299490 p^{72} T^{152} + 56700 p^{74} T^{154} + 200 p^{76} T^{156} - 40 p^{78} T^{158} + p^{80} T^{160}$$
29 $$1 - 280T^{2} + 3.80e4T^{4} - 3.29e6T^{6} + 2.01e8T^{8} - 9.24e9T^{10} + 3.30e11T^{12} - 9.72e12T^{14} + 2.38e14T^{16} - 3.78e15T^{18} - 7.77e16T^{20} + 1.07e19T^{22} - 5.69e20T^{24} + 2.12e22T^{26} - 6.28e23T^{28} + 1.55e25T^{30} - 2.81e26T^{32} - 7.48e25T^{34} + 3.23e29T^{36} - 1.85e31T^{38} + 7.07e32T^{40} - 2.14e34T^{42} + 5.34e35T^{44} - 9.80e36T^{46} + 4.48e37T^{48} + 6.45e39T^{50} - 3.83e41T^{52} + 1.47e43T^{54} - 4.49e44T^{56} + 1.08e46T^{58} - 1.81e47T^{60} + 4.04e47T^{62} + 1.16e50T^{64} - 5.99e51T^{66} + 2.09e53T^{68} - 5.71e54T^{70} + 1.09e56T^{72} - 6.50e56T^{74} - 5.22e58T^{76}+O(T^{77})$$
31 $$1 - 390T^{2} + 720T^{3} + 7.31e4T^{4} - 2.85e5T^{5} - 8.37e6T^{6} + 5.43e7T^{7} + 5.98e8T^{8} - 6.47e9T^{9} - 1.99e10T^{10} + 5.16e11T^{11} - 9.42e11T^{12} - 2.66e13T^{13} + 1.77e14T^{14} + 6.10e14T^{15} - 1.22e16T^{16} + 3.02e16T^{17} + 4.43e17T^{18} - 3.72e18T^{19} - 6.45e17T^{20} + 1.71e20T^{21} - 9.02e20T^{22} - 2.49e21T^{23} + 5.15e22T^{24} - 1.81e23T^{25} - 1.04e24T^{26} + 1.30e25T^{27} - 3.44e25T^{28} - 2.92e26T^{29} + 3.03e27T^{30} - 7.49e27T^{31} - 6.64e28T^{32} + 7.04e29T^{33} - 2.01e30T^{34} - 1.39e31T^{35} + 1.68e32T^{36} - 5.60e32T^{37} - 2.89e33T^{38} + 4.05e34T^{39} - 1.48e35T^{40} - 6.06e35T^{41} + 9.39e36T^{42} - 3.58e37T^{43} - 1.22e38T^{44} + 2.03e39T^{45} - 8.12e39T^{46} - 2.12e40T^{47} + 4.08e41T^{48} - 1.79e42T^{49} - 2.56e42T^{50} + 7.64e43T^{51} - 4.00e44T^{52} - 1.55e43T^{53} + 1.37e46T^{54} - 8.89e46T^{55} + 1.09e47T^{56} + 2.37e48T^{57} - 1.92e49T^{58} + 4.21e49T^{59} + 3.95e50T^{60} - 3.95e51T^{61} + 1.15e52T^{62} + 6.07e52T^{63} - 7.60e53T^{64} + 2.72e54T^{65} + 7.75e54T^{66} - 1.36e56T^{67} + 6.03e56T^{68} + 5.90e56T^{69} - 2.31e58T^{70} + 1.27e59T^{71} - 6.15e58T^{72} - 3.74e60T^{73} + 2.57e61T^{74}+O(T^{75})$$
37 $$1 + 24T + 228T^{2} + 696T^{3} - 5.43e3T^{4} - 4.14e4T^{5} + 3.33e5T^{6} + 5.98e6T^{7} + 2.72e7T^{8} - 3.25e7T^{9} - 1.47e8T^{10} + 1.00e10T^{11} + 9.30e10T^{12} + 9.42e10T^{13} - 2.45e12T^{14} + 1.18e12T^{15} + 2.24e14T^{16} + 1.34e15T^{17} - 1.73e15T^{18} - 3.43e16T^{19} + 1.92e17T^{20} + 3.33e18T^{21} + 7.28e18T^{22} - 9.02e19T^{23} - 3.02e20T^{24} + 4.98e21T^{25} + 3.73e22T^{26} - 5.53e22T^{27} - 1.31e24T^{28} + 2.19e24T^{29} + 8.26e25T^{30} + 2.30e26T^{31} - 2.30e27T^{32} - 1.16e28T^{33} + 9.32e28T^{34} + 8.65e29T^{35} - 9.02e29T^{36} - 3.21e31T^{37} + 3.74e30T^{38} + 1.64e33T^{39} + 5.77e33T^{40} - 4.16e34T^{41} - 2.63e35T^{42} + 1.48e36T^{43} + 1.64e37T^{44} - 8.42e36T^{45} - 5.96e38T^{46} - 4.21e38T^{47} + 2.73e40T^{48} + 1.10e41T^{49} - 6.53e41T^{50} - 4.79e42T^{51} + 2.13e43T^{52} + 2.75e44T^{53} - 1.59e43T^{54} - 9.73e45T^{55} - 1.42e46T^{56} + 3.99e47T^{57} + 1.82e48T^{58} - 9.17e48T^{59} - 7.79e49T^{60} + 2.65e50T^{61} + 4.12e51T^{62} + 1.93e51T^{63} - 1.39e53T^{64} - 2.97e53T^{65} + 5.22e54T^{66} + 2.69e55T^{67} - 1.14e56T^{68} - 1.11e57T^{69} + 2.95e57T^{70} + 5.61e58T^{71}+O(T^{72})$$
41 $$1 + 500T^{2} + 1.30e5T^{4} + 2.38e7T^{6} + 3.45e9T^{8} + 4.20e11T^{10} + 4.46e13T^{12} + 4.23e15T^{14} + 3.65e17T^{16} + 2.90e19T^{18} + 2.14e21T^{20} + 1.49e23T^{22} + 9.78e24T^{24} + 6.10e26T^{26} + 3.63e28T^{28} + 2.08e30T^{30} + 1.15e32T^{32} + 6.16e33T^{34} + 3.21e35T^{36} + 1.64e37T^{38} + 8.19e38T^{40} + 4.01e40T^{42} + 1.94e42T^{44} + 9.23e43T^{46} + 4.33e45T^{48} + 2.00e47T^{50} + 9.20e48T^{52} + 4.16e50T^{54} + 1.86e52T^{56} + 8.30e53T^{58} + 3.66e55T^{60} + 1.60e57T^{62} + 6.96e58T^{64} + 3.00e60T^{66} + 1.28e62T^{68}+O(T^{70})$$
43 $$1 + 24T + 288T^{2} + 1.84e3T^{3} - 632T^{4} - 1.32e5T^{5} - 1.29e6T^{6} - 5.45e6T^{7} + 4.24e6T^{8} + 1.20e8T^{9} - 4.72e8T^{10} - 1.17e10T^{11} - 3.81e10T^{12} + 4.62e11T^{13} + 4.79e12T^{14} + 3.29e12T^{15} - 1.98e14T^{16} - 7.98e14T^{17} + 9.42e15T^{18} + 1.11e17T^{19} + 2.46e17T^{20} - 2.82e18T^{21} - 1.15e19T^{22} + 2.12e20T^{23} + 2.55e21T^{24} + 7.97e21T^{25} - 5.26e22T^{26} - 4.94e23T^{27} + 3.17e23T^{28} + 1.89e25T^{29} - 1.54e25T^{30} - 1.76e27T^{31} - 1.24e28T^{32} + 1.00e27T^{33} + 4.81e29T^{34} + 1.55e30T^{35} - 2.05e31T^{36} - 2.10e32T^{37} - 3.25e32T^{38} + 6.05e33T^{39} + 2.77e34T^{40} - 2.40e35T^{41} - 2.72e36T^{42} - 1.50e36T^{43} + 1.30e38T^{44} + 8.03e38T^{45} - 9.42e38T^{46} - 2.76e40T^{47} + 3.22e40T^{48} + 2.19e42T^{49} + 1.29e43T^{50} - 2.04e43T^{51} - 5.86e44T^{52} - 1.22e45T^{53} + 2.40e46T^{54} + 1.84e47T^{55} - 8.96e46T^{56} - 7.51e48T^{57} - 2.25e49T^{58} + 2.57e50T^{59} + 2.00e51T^{60} - 5.67e51T^{61} - 1.46e53T^{62} - 6.52e53T^{63} + 1.78e54T^{64} + 2.48e55T^{65} - 4.24e55T^{66} - 1.70e57T^{67} - 7.82e57T^{68}+O(T^{69})$$
47 $$1 + 80T^{2} + 8.08e3T^{4} + 2.79e5T^{6} + 1.73e7T^{8} + 1.19e8T^{10} + 1.02e10T^{12} - 1.64e12T^{14} - 4.74e13T^{16} - 7.40e15T^{18} - 1.31e17T^{20} - 1.30e19T^{22} + 5.72e20T^{24} + 1.79e22T^{26} + 3.30e24T^{28} + 8.82e25T^{30} + 7.48e27T^{32} + 7.61e28T^{34} + 5.88e30T^{36} - 5.45e32T^{38} - 1.83e34T^{40} - 2.45e36T^{42} - 5.15e37T^{44} - 3.20e39T^{46} + 7.42e40T^{48} + 2.72e42T^{50} + 5.86e44T^{52} + 2.14e46T^{54} + 1.66e48T^{56} + 3.06e49T^{58} + 1.82e51T^{60} - 6.78e52T^{62} - 2.47e54T^{64} - 3.84e56T^{66}+O(T^{67})$$
53 $$1 + 420T^{2} + 8.57e4T^{4} + 1.27e7T^{6} + 1.68e9T^{8} + 2.06e11T^{10} + 2.29e13T^{12} + 2.35e15T^{14} + 2.29e17T^{16} + 2.12e19T^{18} + 1.87e21T^{20} + 1.59e23T^{22} + 1.30e25T^{24} + 1.03e27T^{26} + 7.96e28T^{28} + 5.96e30T^{30} + 4.36e32T^{32} + 3.11e34T^{34} + 2.18e36T^{36} + 1.49e38T^{38} + 1.01e40T^{40} + 6.69e41T^{42} + 4.37e43T^{44} + 2.81e45T^{46} + 1.78e47T^{48} + 1.11e49T^{50} + 6.88e50T^{52} + 4.19e52T^{54} + 2.53e54T^{56} + 1.50e56T^{58} + 8.88e57T^{60} + 5.17e59T^{62} + 2.98e61T^{64}+O(T^{65})$$
59 $$1 - 680T^{2} + 2.37e5T^{4} - 5.66e7T^{6} + 1.04e10T^{8} - 1.57e12T^{10} + 2.04e14T^{12} - 2.32e16T^{14} + 2.38e18T^{16} - 2.23e20T^{18} + 1.94e22T^{20} - 1.60e24T^{22} + 1.26e26T^{24} - 9.64e27T^{26} + 7.20e29T^{28} - 5.30e31T^{30} + 3.84e33T^{32} - 2.76e35T^{34} + 1.95e37T^{36} - 1.37e39T^{38} + 9.49e40T^{40} - 6.50e42T^{42} + 4.41e44T^{44} - 2.96e46T^{46} + 1.96e48T^{48} - 1.29e50T^{50} + 8.43e51T^{52} - 5.46e53T^{54} + 3.50e55T^{56} - 2.23e57T^{58} + 1.41e59T^{60} - 8.92e60T^{62}+O(T^{64})$$
61 $$1 - 480T^{2} + 1.04e3T^{3} + 1.21e5T^{4} - 4.93e5T^{5} - 2.05e7T^{6} + 1.22e8T^{7} + 2.51e9T^{8} - 2.06e10T^{9} - 2.24e11T^{10} + 2.59e12T^{11} + 1.32e13T^{12} - 2.54e14T^{13} - 2.08e14T^{14} + 1.96e16T^{15} - 6.57e16T^{16} - 1.16e18T^{17} + 1.03e19T^{18} + 4.36e19T^{19} - 9.62e20T^{20} + 3.50e20T^{21} + 6.37e22T^{22} - 2.38e23T^{23} - 2.69e24T^{24} + 2.54e25T^{25} - 5.33e24T^{26} - 1.80e27T^{27} + 1.44e28T^{28} + 8.82e28T^{29} - 1.69e30T^{30} - 1.54e30T^{31} + 1.30e32T^{32} - 2.81e32T^{33} - 7.35e33T^{34} + 4.59e34T^{35} + 2.56e35T^{36} - 4.54e36T^{37} + 2.94e36T^{38} + 3.47e38T^{39} - 1.39e39T^{40} - 2.07e40T^{41} + 1.39e41T^{42} + 8.44e41T^{43} - 9.37e42T^{44} - 3.31e42T^{45} + 4.52e44T^{46} - 3.40e45T^{47} - 1.15e46T^{48} + 3.91e47T^{49} - 5.67e47T^{50} - 2.78e49T^{51} + 1.12e50T^{52} + 1.31e51T^{53} - 1.07e52T^{54} - 2.03e52T^{55} + 7.99e53T^{56} - 3.49e54T^{57} - 4.95e55T^{58} + 4.66e56T^{59} + 2.46e57T^{60} - 3.59e58T^{61} - 7.72e58T^{62}+O(T^{63})$$
67 $$1 - 96T + 5.32e3T^{2} - 2.18e5T^{3} + 7.34e6T^{4} - 2.11e8T^{5} + 5.36e9T^{6} - 1.22e11T^{7} + 2.57e12T^{8} - 4.98e13T^{9} + 8.96e14T^{10} - 1.51e16T^{11} + 2.38e17T^{12} - 3.56e18T^{13} + 5.01e19T^{14} - 6.68e20T^{15} + 8.40e21T^{16} - 9.97e22T^{17} + 1.11e24T^{18} - 1.15e25T^{19} + 1.11e26T^{20} - 9.68e26T^{21} + 7.23e27T^{22} - 3.98e28T^{23} + 3.23e28T^{24} + 3.27e30T^{25} - 6.30e31T^{26} + 8.30e32T^{27} - 8.97e33T^{28} + 8.23e34T^{29} - 6.23e35T^{30} + 3.36e36T^{31} - 1.15e36T^{32} - 2.93e38T^{33} + 5.28e39T^{34} - 6.58e40T^{35} + 6.71e41T^{36} - 5.74e42T^{37} + 3.96e43T^{38} - 1.76e44T^{39} - 4.28e44T^{40} + 2.23e46T^{41} - 3.38e47T^{42} + 3.79e48T^{43} - 3.48e49T^{44} + 2.64e50T^{45} - 1.49e51T^{46} + 3.05e51T^{47} + 7.11e52T^{48} - 1.40e54T^{49} + 1.71e55T^{50} - 1.66e56T^{51} + 1.32e57T^{52} - 8.31e57T^{53} + 3.00e58T^{54} + 1.63e59T^{55} - 4.90e60T^{56} + 6.53e61T^{57} - 6.60e62T^{58} + 5.48e63T^{59} - 3.65e64T^{60} + 1.64e65T^{61}+O(T^{62})$$
71 $$1 + 1.28e3T^{2} + 8.11e5T^{4} + 3.40e8T^{6} + 1.06e11T^{8} + 2.67e13T^{10} + 5.58e15T^{12} + 1.00e18T^{14} + 1.60e20T^{16} + 2.31e22T^{18} + 3.06e24T^{20} + 3.79e26T^{22} + 4.41e28T^{24} + 4.89e30T^{26} + 5.19e32T^{28} + 5.32e34T^{30} + 5.28e36T^{32} + 5.10e38T^{34} + 4.81e40T^{36} + 4.45e42T^{38} + 4.03e44T^{40} + 3.59e46T^{42} + 3.14e48T^{44} + 2.70e50T^{46} + 2.30e52T^{48} + 1.92e54T^{50} + 1.59e56T^{52} + 1.29e58T^{54} + 1.04e60T^{56} + 8.35e61T^{58} + 6.58e63T^{60}+O(T^{61})$$
73 $$1 - 100T + 5.60e3T^{2} - 2.27e5T^{3} + 7.46e6T^{4} - 2.09e8T^{5} + 5.16e9T^{6} - 1.15e11T^{7} + 2.37e12T^{8} - 4.52e13T^{9} + 8.09e14T^{10} - 1.36e16T^{11} + 2.19e17T^{12} - 3.37e18T^{13} + 4.96e19T^{14} - 7.03e20T^{15} + 9.62e21T^{16} - 1.27e23T^{17} + 1.63e24T^{18} - 2.05e25T^{19} + 2.50e26T^{20} - 2.99e27T^{21} + 3.51e28T^{22} - 4.03e29T^{23} + 4.56e30T^{24} - 5.08e31T^{25} + 5.58e32T^{26} - 6.06e33T^{27} + 6.50e34T^{28} - 6.91e35T^{29} + 7.28e36T^{30} - 7.59e37T^{31} + 7.86e38T^{32} - 8.08e39T^{33} + 8.24e40T^{34} - 8.34e41T^{35} + 8.40e42T^{36} - 8.40e43T^{37} + 8.35e44T^{38} - 8.25e45T^{39} + 8.11e46T^{40} - 7.94e47T^{41} + 7.72e48T^{42} - 7.48e49T^{43} + 7.21e50T^{44} - 6.92e51T^{45} + 6.61e52T^{46} - 6.28e53T^{47} + 5.95e54T^{48} - 5.61e55T^{49} + 5.27e56T^{50} - 4.94e57T^{51} + 4.60e58T^{52} - 4.28e59T^{53} + 3.96e60T^{54} - 3.65e61T^{55} + 3.36e62T^{56} - 3.08e63T^{57} + 2.81e64T^{58} - 2.56e65T^{59} + 2.32e66T^{60}+O(T^{61})$$
79 $$1 - 80T + 3.90e3T^{2} - 1.45e5T^{3} + 4.50e6T^{4} - 1.22e8T^{5} + 2.97e9T^{6} - 6.64e10T^{7} + 1.37e12T^{8} - 2.68e13T^{9} + 4.93e14T^{10} - 8.64e15T^{11} + 1.44e17T^{12} - 2.32e18T^{13} + 3.61e19T^{14} - 5.41e20T^{15} + 7.86e21T^{16} - 1.10e23T^{17} + 1.52e24T^{18} - 2.03e25T^{19} + 2.65e26T^{20} - 3.37e27T^{21} + 4.20e28T^{22} - 5.11e29T^{23} + 6.09e30T^{24} - 7.11e31T^{25} + 8.13e32T^{26} - 9.12e33T^{27} + 1.00e35T^{28} - 1.07e36T^{29} + 1.13e37T^{30} - 1.17e38T^{31} + 1.19e39T^{32} - 1.18e40T^{33} + 1.14e41T^{34} - 1.08e42T^{35} + 9.94e42T^{36} - 8.86e43T^{37} + 7.61e44T^{38} - 6.24e45T^{39} + 4.81e46T^{40} - 3.37e47T^{41} + 1.97e48T^{42} - 6.72e48T^{43} - 5.01e49T^{44} + 1.51e51T^{45} - 2.34e52T^{46} + 2.98e53T^{47} - 3.44e54T^{48} + 3.71e55T^{49} - 3.83e56T^{50} + 3.81e57T^{51} - 3.69e58T^{52} + 3.47e59T^{53} - 3.20e60T^{54} + 2.90e61T^{55} - 2.58e62T^{56} + 2.26e63T^{57} - 1.95e64T^{58} + 1.66e65T^{59}+O(T^{60})$$
83 $$1 - 920T^{2} + 3.80e5T^{4} - 9.05e7T^{6} + 1.26e10T^{8} - 7.65e11T^{10} - 6.88e13T^{12} + 1.95e16T^{14} - 1.64e18T^{16} - 1.01e19T^{18} + 1.66e22T^{20} - 1.80e24T^{22} + 8.46e25T^{24} + 1.93e27T^{26} - 8.60e29T^{28} + 1.12e32T^{30} - 8.18e33T^{32} - 6.41e34T^{34} + 8.87e37T^{36} - 9.48e39T^{38} + 3.39e41T^{40} + 2.45e43T^{42} - 4.12e45T^{44} + 3.24e47T^{46} - 1.94e49T^{48} + 1.64e50T^{50} + 1.90e53T^{52} - 2.61e55T^{54} + 1.30e57T^{56} + 4.16e58T^{58}+O(T^{59})$$
89 $$1 - 2.04e3T^{2} + 2.00e6T^{4} - 1.26e9T^{6} + 5.68e11T^{8} - 1.92e14T^{10} + 5.04e16T^{12} - 1.02e19T^{14} + 1.54e21T^{16} - 1.54e23T^{18} + 3.87e24T^{20} + 1.97e27T^{22} - 4.26e29T^{24} + 4.27e31T^{26} - 7.08e32T^{28} - 4.78e35T^{30} + 7.80e37T^{32} - 4.78e39T^{34} - 2.97e41T^{36} + 9.22e43T^{38} - 8.06e45T^{40} - 7.80e45T^{42} + 8.14e49T^{44} - 8.96e51T^{46} + 2.17e53T^{48} + 5.56e55T^{50} - 7.37e57T^{52} + 2.62e59T^{54} + 3.22e61T^{56}+O(T^{58})$$
97 $$1 + 32T + 902T^{2} + 1.76e4T^{3} + 3.21e5T^{4} + 5.08e6T^{5} + 7.38e7T^{6} + 9.87e8T^{7} + 1.22e10T^{8} + 1.48e11T^{9} + 1.70e12T^{10} + 1.92e13T^{11} + 2.12e14T^{12} + 2.29e15T^{13} + 2.40e16T^{14} + 2.37e17T^{15} + 2.23e18T^{16} + 1.94e19T^{17} + 1.57e20T^{18} + 1.20e21T^{19} + 8.48e21T^{20} + 6.28e22T^{21} + 4.76e23T^{22} + 3.30e24T^{23} + 1.93e25T^{24} + 4.41e24T^{25} - 1.18e27T^{26} - 1.40e28T^{27} - 1.10e28T^{28} + 3.70e30T^{29} + 7.75e31T^{30} + 1.11e33T^{31} + 1.21e34T^{32} + 9.68e34T^{33} + 5.70e35T^{34} + 1.24e35T^{35} - 4.07e37T^{36} - 6.56e38T^{37} - 8.70e39T^{38} - 1.11e41T^{39} - 1.71e42T^{40} - 2.56e43T^{41} - 3.53e44T^{42} - 4.43e45T^{43} - 4.83e46T^{44} - 4.78e47T^{45} - 4.33e48T^{46} - 3.78e49T^{47} - 3.51e50T^{48} - 3.50e51T^{49} - 3.72e52T^{50} - 3.83e53T^{51} - 3.44e54T^{52} - 2.64e55T^{53} - 1.44e56T^{54} - 3.58e56T^{55} + 2.12e57T^{56}+O(T^{57})$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{160} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}