# Properties

 Degree 64 Conductor $2^{32} \cdot 3^{32} \cdot 5^{32} \cdot 7^{32}$ Sign $1$ Motivic weight 2 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 16·2-s + 128·4-s − 8·7-s − 704·8-s − 32·11-s − 32·13-s + 128·14-s + 3.08e3·16-s − 56·17-s + 512·22-s − 48·23-s + 34·25-s + 512·26-s − 1.02e3·28-s + 160·31-s − 1.17e4·32-s + 896·34-s + 44·37-s − 80·41-s − 184·43-s − 4.09e3·44-s + 768·46-s − 228·47-s + 32·49-s − 544·50-s − 4.09e3·52-s + 48·53-s + ⋯
 L(s)  = 1 − 8·2-s + 32·4-s − 8/7·7-s − 88·8-s − 2.90·11-s − 2.46·13-s + 64/7·14-s + 193·16-s − 3.29·17-s + 23.2·22-s − 2.08·23-s + 1.35·25-s + 19.6·26-s − 36.5·28-s + 5.16·31-s − 366·32-s + 26.3·34-s + 1.18·37-s − 1.95·41-s − 4.27·43-s − 93.0·44-s + 16.6·46-s − 4.85·47-s + 0.653·49-s − 10.8·50-s − 78.7·52-s + 0.905·53-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 5^{32} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{32} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 5^{32} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s+1)^{32} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 $$d$$ = $$64$$ $$N$$ = $$2^{32} \cdot 3^{32} \cdot 5^{32} \cdot 7^{32}$$ $$\varepsilon$$ = $1$ motivic weight = $$2$$ character : induced by $\chi_{210} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$0$$ Selberg data = $$(64,\ 2^{32} \cdot 3^{32} \cdot 5^{32} \cdot 7^{32} ,\ ( \ : [1]^{32} ),\ 1 )$$ $$L(\frac{3}{2})$$ $$\approx$$ $$0.119837$$ $$L(\frac12)$$ $$\approx$$ $$0.119837$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;5,\;7\}$,$$F_p(T)$$ is a polynomial of degree 64. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 63.
$p$$F_p(T)$
bad2 $$( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} )^{8}$$
3 $$( 1 - p^{2} T^{4} + p^{4} T^{8} )^{4}$$
5 $$1 - 34 T^{2} - 336 T^{3} - 676 T^{4} + 10752 T^{5} + 87436 T^{6} + 170856 T^{7} - 1241746 T^{8} - 2894304 p T^{9} - 1752182 p^{2} T^{10} + 3560592 p^{2} T^{11} + 11472552 p^{3} T^{12} + 12160008 p^{4} T^{13} + 11630394 p^{4} T^{14} - 40712448 p^{5} T^{15} - 57899153 p^{6} T^{16} - 40712448 p^{7} T^{17} + 11630394 p^{8} T^{18} + 12160008 p^{10} T^{19} + 11472552 p^{11} T^{20} + 3560592 p^{12} T^{21} - 1752182 p^{14} T^{22} - 2894304 p^{15} T^{23} - 1241746 p^{16} T^{24} + 170856 p^{18} T^{25} + 87436 p^{20} T^{26} + 10752 p^{22} T^{27} - 676 p^{24} T^{28} - 336 p^{26} T^{29} - 34 p^{28} T^{30} + p^{32} T^{32}$$
7 $$1 + 8 T + 32 T^{2} - 428 T^{3} - 2340 T^{4} - 6108 T^{5} + 117608 T^{6} - 188852 T^{7} + 261090 T^{8} - 1664636 p T^{9} + 5573688 p^{2} T^{10} - 2426336 p^{3} T^{11} - 3931244 p^{4} T^{12} - 6019560 p^{5} T^{13} + 4643080 p^{6} T^{14} + 4813420 p^{7} T^{15} + 9246675 p^{8} T^{16} + 4813420 p^{9} T^{17} + 4643080 p^{10} T^{18} - 6019560 p^{11} T^{19} - 3931244 p^{12} T^{20} - 2426336 p^{13} T^{21} + 5573688 p^{14} T^{22} - 1664636 p^{15} T^{23} + 261090 p^{16} T^{24} - 188852 p^{18} T^{25} + 117608 p^{20} T^{26} - 6108 p^{22} T^{27} - 2340 p^{24} T^{28} - 428 p^{26} T^{29} + 32 p^{28} T^{30} + 8 p^{30} T^{31} + p^{32} T^{32}$$
good11 $$( 1 + 16 T - 322 T^{2} - 2984 T^{3} + 84470 T^{4} - 50260 T^{5} - 16142276 T^{6} + 91798084 T^{7} + 2040053636 T^{8} - 1281794744 p T^{9} - 13867581876 p T^{10} + 1166843885404 T^{11} + 10952808020900 T^{12} - 73850587314660 T^{13} - 2325064981249694 T^{14} + 2308470374815584 T^{15} + 380031413761461427 T^{16} + 2308470374815584 p^{2} T^{17} - 2325064981249694 p^{4} T^{18} - 73850587314660 p^{6} T^{19} + 10952808020900 p^{8} T^{20} + 1166843885404 p^{10} T^{21} - 13867581876 p^{13} T^{22} - 1281794744 p^{15} T^{23} + 2040053636 p^{16} T^{24} + 91798084 p^{18} T^{25} - 16142276 p^{20} T^{26} - 50260 p^{22} T^{27} + 84470 p^{24} T^{28} - 2984 p^{26} T^{29} - 322 p^{28} T^{30} + 16 p^{30} T^{31} + p^{32} T^{32} )^{2}$$
13 $$( 1 + 16 T + 128 T^{2} + 188 p T^{3} + 14316 T^{4} - 343380 T^{5} - 4339960 T^{6} - 110487700 T^{7} - 2516638446 T^{8} - 19922152156 T^{9} - 145570872648 T^{10} - 2758835887024 T^{11} - 409842977996 T^{12} + 496418351850600 T^{13} + 31710506382280 p^{2} T^{14} + 8059325024212700 p T^{15} + 2142629679076020195 T^{16} + 8059325024212700 p^{3} T^{17} + 31710506382280 p^{6} T^{18} + 496418351850600 p^{6} T^{19} - 409842977996 p^{8} T^{20} - 2758835887024 p^{10} T^{21} - 145570872648 p^{12} T^{22} - 19922152156 p^{14} T^{23} - 2516638446 p^{16} T^{24} - 110487700 p^{18} T^{25} - 4339960 p^{20} T^{26} - 343380 p^{22} T^{27} + 14316 p^{24} T^{28} + 188 p^{27} T^{29} + 128 p^{28} T^{30} + 16 p^{30} T^{31} + p^{32} T^{32} )^{2}$$
17 $$1 + 56 T + 1568 T^{2} + 21560 T^{3} - 40112 T^{4} - 8811572 T^{5} - 198135616 T^{6} - 1982747424 T^{7} + 11700324696 T^{8} + 993772764656 T^{9} + 25431415962568 T^{10} + 421993637314564 T^{11} + 3651160992412552 T^{12} - 39831914850219044 T^{13} - 2310125442796729152 T^{14} - 49580163585504631296 T^{15} -$$$$56\!\cdots\!70$$$$T^{16} +$$$$98\!\cdots\!64$$$$T^{17} +$$$$21\!\cdots\!64$$$$T^{18} +$$$$52\!\cdots\!12$$$$T^{19} +$$$$71\!\cdots\!56$$$$T^{20} +$$$$27\!\cdots\!64$$$$T^{21} -$$$$15\!\cdots\!20$$$$T^{22} -$$$$46\!\cdots\!36$$$$T^{23} -$$$$75\!\cdots\!52$$$$T^{24} -$$$$59\!\cdots\!88$$$$T^{25} +$$$$59\!\cdots\!36$$$$T^{26} +$$$$30\!\cdots\!24$$$$T^{27} +$$$$59\!\cdots\!00$$$$T^{28} +$$$$60\!\cdots\!04$$$$T^{29} -$$$$27\!\cdots\!48$$$$T^{30} -$$$$26\!\cdots\!84$$$$T^{31} -$$$$60\!\cdots\!01$$$$T^{32} -$$$$26\!\cdots\!84$$$$p^{2} T^{33} -$$$$27\!\cdots\!48$$$$p^{4} T^{34} +$$$$60\!\cdots\!04$$$$p^{6} T^{35} +$$$$59\!\cdots\!00$$$$p^{8} T^{36} +$$$$30\!\cdots\!24$$$$p^{10} T^{37} +$$$$59\!\cdots\!36$$$$p^{12} T^{38} -$$$$59\!\cdots\!88$$$$p^{14} T^{39} -$$$$75\!\cdots\!52$$$$p^{16} T^{40} -$$$$46\!\cdots\!36$$$$p^{18} T^{41} -$$$$15\!\cdots\!20$$$$p^{20} T^{42} +$$$$27\!\cdots\!64$$$$p^{22} T^{43} +$$$$71\!\cdots\!56$$$$p^{24} T^{44} +$$$$52\!\cdots\!12$$$$p^{26} T^{45} +$$$$21\!\cdots\!64$$$$p^{28} T^{46} +$$$$98\!\cdots\!64$$$$p^{30} T^{47} -$$$$56\!\cdots\!70$$$$p^{32} T^{48} - 49580163585504631296 p^{34} T^{49} - 2310125442796729152 p^{36} T^{50} - 39831914850219044 p^{38} T^{51} + 3651160992412552 p^{40} T^{52} + 421993637314564 p^{42} T^{53} + 25431415962568 p^{44} T^{54} + 993772764656 p^{46} T^{55} + 11700324696 p^{48} T^{56} - 1982747424 p^{50} T^{57} - 198135616 p^{52} T^{58} - 8811572 p^{54} T^{59} - 40112 p^{56} T^{60} + 21560 p^{58} T^{61} + 1568 p^{60} T^{62} + 56 p^{62} T^{63} + p^{64} T^{64}$$
19 $$1 + 2344 T^{2} + 3033068 T^{4} + 2502523792 T^{6} + 1346852178486 T^{8} + 339036813313568 T^{10} - 143875083497505416 T^{12} -$$$$21\!\cdots\!56$$$$T^{14} -$$$$12\!\cdots\!23$$$$T^{16} -$$$$35\!\cdots\!40$$$$T^{18} +$$$$26\!\cdots\!76$$$$T^{20} +$$$$88\!\cdots\!52$$$$T^{22} +$$$$49\!\cdots\!30$$$$T^{24} +$$$$13\!\cdots\!84$$$$T^{26} -$$$$16\!\cdots\!76$$$$T^{28} -$$$$59\!\cdots\!48$$$$p^{2} T^{30} -$$$$84\!\cdots\!80$$$$p^{4} T^{32} -$$$$59\!\cdots\!48$$$$p^{6} T^{34} -$$$$16\!\cdots\!76$$$$p^{8} T^{36} +$$$$13\!\cdots\!84$$$$p^{12} T^{38} +$$$$49\!\cdots\!30$$$$p^{16} T^{40} +$$$$88\!\cdots\!52$$$$p^{20} T^{42} +$$$$26\!\cdots\!76$$$$p^{24} T^{44} -$$$$35\!\cdots\!40$$$$p^{28} T^{46} -$$$$12\!\cdots\!23$$$$p^{32} T^{48} -$$$$21\!\cdots\!56$$$$p^{36} T^{50} - 143875083497505416 p^{40} T^{52} + 339036813313568 p^{44} T^{54} + 1346852178486 p^{48} T^{56} + 2502523792 p^{52} T^{58} + 3033068 p^{56} T^{60} + 2344 p^{60} T^{62} + p^{64} T^{64}$$
23 $$1 + 48 T + 1152 T^{2} - 7592 T^{3} - 672304 T^{4} - 13448884 T^{5} + 157767008 T^{6} + 5516634208 T^{7} - 53384392952 T^{8} - 6794784279760 T^{9} - 110029480685752 T^{10} + 2462050691124260 T^{11} + 166901127855959144 T^{12} + 173435548151687172 p T^{13} + 937338481100674720 p T^{14} -$$$$14\!\cdots\!48$$$$T^{15} -$$$$32\!\cdots\!26$$$$T^{16} +$$$$11\!\cdots\!68$$$$T^{17} +$$$$18\!\cdots\!92$$$$T^{18} +$$$$56\!\cdots\!24$$$$T^{19} -$$$$13\!\cdots\!28$$$$T^{20} -$$$$42\!\cdots\!68$$$$T^{21} -$$$$13\!\cdots\!88$$$$T^{22} +$$$$21\!\cdots\!64$$$$T^{23} +$$$$60\!\cdots\!52$$$$T^{24} +$$$$54\!\cdots\!32$$$$T^{25} -$$$$12\!\cdots\!80$$$$T^{26} -$$$$32\!\cdots\!40$$$$T^{27} +$$$$42\!\cdots\!72$$$$T^{28} +$$$$32\!\cdots\!28$$$$T^{29} +$$$$41\!\cdots\!28$$$$T^{30} -$$$$12\!\cdots\!16$$$$T^{31} -$$$$53\!\cdots\!37$$$$T^{32} -$$$$12\!\cdots\!16$$$$p^{2} T^{33} +$$$$41\!\cdots\!28$$$$p^{4} T^{34} +$$$$32\!\cdots\!28$$$$p^{6} T^{35} +$$$$42\!\cdots\!72$$$$p^{8} T^{36} -$$$$32\!\cdots\!40$$$$p^{10} T^{37} -$$$$12\!\cdots\!80$$$$p^{12} T^{38} +$$$$54\!\cdots\!32$$$$p^{14} T^{39} +$$$$60\!\cdots\!52$$$$p^{16} T^{40} +$$$$21\!\cdots\!64$$$$p^{18} T^{41} -$$$$13\!\cdots\!88$$$$p^{20} T^{42} -$$$$42\!\cdots\!68$$$$p^{22} T^{43} -$$$$13\!\cdots\!28$$$$p^{24} T^{44} +$$$$56\!\cdots\!24$$$$p^{26} T^{45} +$$$$18\!\cdots\!92$$$$p^{28} T^{46} +$$$$11\!\cdots\!68$$$$p^{30} T^{47} -$$$$32\!\cdots\!26$$$$p^{32} T^{48} -$$$$14\!\cdots\!48$$$$p^{34} T^{49} + 937338481100674720 p^{37} T^{50} + 173435548151687172 p^{39} T^{51} + 166901127855959144 p^{40} T^{52} + 2462050691124260 p^{42} T^{53} - 110029480685752 p^{44} T^{54} - 6794784279760 p^{46} T^{55} - 53384392952 p^{48} T^{56} + 5516634208 p^{50} T^{57} + 157767008 p^{52} T^{58} - 13448884 p^{54} T^{59} - 672304 p^{56} T^{60} - 7592 p^{58} T^{61} + 1152 p^{60} T^{62} + 48 p^{62} T^{63} + p^{64} T^{64}$$
29 $$( 1 - 6720 T^{2} + 24117516 T^{4} - 60007636444 T^{6} + 114277499004756 T^{8} - 175249552287471988 T^{10} +$$$$22\!\cdots\!16$$$$T^{12} -$$$$23\!\cdots\!68$$$$T^{14} +$$$$21\!\cdots\!22$$$$T^{16} -$$$$23\!\cdots\!68$$$$p^{4} T^{18} +$$$$22\!\cdots\!16$$$$p^{8} T^{20} - 175249552287471988 p^{12} T^{22} + 114277499004756 p^{16} T^{24} - 60007636444 p^{20} T^{26} + 24117516 p^{24} T^{28} - 6720 p^{28} T^{30} + p^{32} T^{32} )^{2}$$
31 $$( 1 - 80 T + 1148 T^{2} + 48304 T^{3} - 1725690 T^{4} + 68427656 T^{5} - 1608024296 T^{6} - 65723247928 T^{7} + 81055630271 p T^{8} - 201838430704 T^{9} - 372303478086952 T^{10} + 7851188509333696 T^{11} + 809862074358801078 T^{12} - 70983157750195402520 T^{13} +$$$$38\!\cdots\!84$$$$T^{14} +$$$$63\!\cdots\!92$$$$T^{15} -$$$$24\!\cdots\!20$$$$T^{16} +$$$$63\!\cdots\!92$$$$p^{2} T^{17} +$$$$38\!\cdots\!84$$$$p^{4} T^{18} - 70983157750195402520 p^{6} T^{19} + 809862074358801078 p^{8} T^{20} + 7851188509333696 p^{10} T^{21} - 372303478086952 p^{12} T^{22} - 201838430704 p^{14} T^{23} + 81055630271 p^{17} T^{24} - 65723247928 p^{18} T^{25} - 1608024296 p^{20} T^{26} + 68427656 p^{22} T^{27} - 1725690 p^{24} T^{28} + 48304 p^{26} T^{29} + 1148 p^{28} T^{30} - 80 p^{30} T^{31} + p^{32} T^{32} )^{2}$$
37 $$1 - 44 T + 968 T^{2} + 80680 T^{3} - 6261832 T^{4} + 219964744 T^{5} - 362364160 T^{6} - 343125693056 T^{7} + 19932719523342 T^{8} - 15617733316640 p T^{9} + 370597233888432 T^{10} + 856798318641781236 T^{11} - 47401629410218598568 T^{12} +$$$$13\!\cdots\!08$$$$T^{13} -$$$$33\!\cdots\!56$$$$T^{14} -$$$$17\!\cdots\!20$$$$T^{15} +$$$$92\!\cdots\!29$$$$T^{16} -$$$$23\!\cdots\!28$$$$T^{17} -$$$$37\!\cdots\!20$$$$T^{18} +$$$$33\!\cdots\!72$$$$T^{19} -$$$$15\!\cdots\!48$$$$T^{20} +$$$$47\!\cdots\!76$$$$T^{21} -$$$$12\!\cdots\!96$$$$T^{22} -$$$$62\!\cdots\!12$$$$T^{23} +$$$$41\!\cdots\!74$$$$T^{24} -$$$$12\!\cdots\!52$$$$T^{25} +$$$$30\!\cdots\!12$$$$T^{26} +$$$$18\!\cdots\!04$$$$T^{27} -$$$$10\!\cdots\!76$$$$T^{28} +$$$$26\!\cdots\!96$$$$T^{29} +$$$$23\!\cdots\!92$$$$T^{30} -$$$$39\!\cdots\!08$$$$T^{31} +$$$$20\!\cdots\!00$$$$T^{32} -$$$$39\!\cdots\!08$$$$p^{2} T^{33} +$$$$23\!\cdots\!92$$$$p^{4} T^{34} +$$$$26\!\cdots\!96$$$$p^{6} T^{35} -$$$$10\!\cdots\!76$$$$p^{8} T^{36} +$$$$18\!\cdots\!04$$$$p^{10} T^{37} +$$$$30\!\cdots\!12$$$$p^{12} T^{38} -$$$$12\!\cdots\!52$$$$p^{14} T^{39} +$$$$41\!\cdots\!74$$$$p^{16} T^{40} -$$$$62\!\cdots\!12$$$$p^{18} T^{41} -$$$$12\!\cdots\!96$$$$p^{20} T^{42} +$$$$47\!\cdots\!76$$$$p^{22} T^{43} -$$$$15\!\cdots\!48$$$$p^{24} T^{44} +$$$$33\!\cdots\!72$$$$p^{26} T^{45} -$$$$37\!\cdots\!20$$$$p^{28} T^{46} -$$$$23\!\cdots\!28$$$$p^{30} T^{47} +$$$$92\!\cdots\!29$$$$p^{32} T^{48} -$$$$17\!\cdots\!20$$$$p^{34} T^{49} -$$$$33\!\cdots\!56$$$$p^{36} T^{50} +$$$$13\!\cdots\!08$$$$p^{38} T^{51} - 47401629410218598568 p^{40} T^{52} + 856798318641781236 p^{42} T^{53} + 370597233888432 p^{44} T^{54} - 15617733316640 p^{47} T^{55} + 19932719523342 p^{48} T^{56} - 343125693056 p^{50} T^{57} - 362364160 p^{52} T^{58} + 219964744 p^{54} T^{59} - 6261832 p^{56} T^{60} + 80680 p^{58} T^{61} + 968 p^{60} T^{62} - 44 p^{62} T^{63} + p^{64} T^{64}$$
41 $$( 1 + 20 T + 1278 T^{2} - 38276 T^{3} + 853934 T^{4} + 4541436 T^{5} + 5875241188 T^{6} - 27274394992 T^{7} + 4538409861312 T^{8} - 27274394992 p^{2} T^{9} + 5875241188 p^{4} T^{10} + 4541436 p^{6} T^{11} + 853934 p^{8} T^{12} - 38276 p^{10} T^{13} + 1278 p^{12} T^{14} + 20 p^{14} T^{15} + p^{16} T^{16} )^{4}$$
43 $$( 1 + 92 T + 4232 T^{2} + 234308 T^{3} + 17156192 T^{4} + 850806472 T^{5} + 33119310312 T^{6} + 1548360435128 T^{7} + 57744913080946 T^{8} + 1516830516564260 T^{9} + 51699751977611360 T^{10} + 1998662968759146740 T^{11} + 4648074954869677772 T^{12} -$$$$25\!\cdots\!36$$$$T^{13} -$$$$99\!\cdots\!56$$$$T^{14} -$$$$44\!\cdots\!64$$$$T^{15} -$$$$19\!\cdots\!41$$$$T^{16} -$$$$44\!\cdots\!64$$$$p^{2} T^{17} -$$$$99\!\cdots\!56$$$$p^{4} T^{18} -$$$$25\!\cdots\!36$$$$p^{6} T^{19} + 4648074954869677772 p^{8} T^{20} + 1998662968759146740 p^{10} T^{21} + 51699751977611360 p^{12} T^{22} + 1516830516564260 p^{14} T^{23} + 57744913080946 p^{16} T^{24} + 1548360435128 p^{18} T^{25} + 33119310312 p^{20} T^{26} + 850806472 p^{22} T^{27} + 17156192 p^{24} T^{28} + 234308 p^{26} T^{29} + 4232 p^{28} T^{30} + 92 p^{30} T^{31} + p^{32} T^{32} )^{2}$$
47 $$1 + 228 T + 25992 T^{2} + 1929672 T^{3} + 102499780 T^{4} + 3910376532 T^{5} + 89208581328 T^{6} - 713393385972 T^{7} - 179847461096628 T^{8} - 9445294864601736 T^{9} - 353884911424936344 T^{10} - 14505371407282045524 T^{11} -$$$$46\!\cdots\!44$$$$T^{12} +$$$$25\!\cdots\!84$$$$T^{13} +$$$$54\!\cdots\!16$$$$T^{14} +$$$$45\!\cdots\!96$$$$T^{15} +$$$$24\!\cdots\!38$$$$T^{16} +$$$$88\!\cdots\!00$$$$T^{17} +$$$$10\!\cdots\!00$$$$T^{18} -$$$$10\!\cdots\!40$$$$T^{19} -$$$$83\!\cdots\!72$$$$T^{20} -$$$$31\!\cdots\!16$$$$T^{21} -$$$$41\!\cdots\!04$$$$T^{22} +$$$$24\!\cdots\!16$$$$T^{23} +$$$$23\!\cdots\!12$$$$T^{24} +$$$$15\!\cdots\!72$$$$T^{25} +$$$$10\!\cdots\!68$$$$T^{26} +$$$$64\!\cdots\!68$$$$T^{27} +$$$$31\!\cdots\!60$$$$T^{28} +$$$$10\!\cdots\!08$$$$T^{29} +$$$$62\!\cdots\!32$$$$T^{30} -$$$$18\!\cdots\!68$$$$T^{31} -$$$$13\!\cdots\!93$$$$T^{32} -$$$$18\!\cdots\!68$$$$p^{2} T^{33} +$$$$62\!\cdots\!32$$$$p^{4} T^{34} +$$$$10\!\cdots\!08$$$$p^{6} T^{35} +$$$$31\!\cdots\!60$$$$p^{8} T^{36} +$$$$64\!\cdots\!68$$$$p^{10} T^{37} +$$$$10\!\cdots\!68$$$$p^{12} T^{38} +$$$$15\!\cdots\!72$$$$p^{14} T^{39} +$$$$23\!\cdots\!12$$$$p^{16} T^{40} +$$$$24\!\cdots\!16$$$$p^{18} T^{41} -$$$$41\!\cdots\!04$$$$p^{20} T^{42} -$$$$31\!\cdots\!16$$$$p^{22} T^{43} -$$$$83\!\cdots\!72$$$$p^{24} T^{44} -$$$$10\!\cdots\!40$$$$p^{26} T^{45} +$$$$10\!\cdots\!00$$$$p^{28} T^{46} +$$$$88\!\cdots\!00$$$$p^{30} T^{47} +$$$$24\!\cdots\!38$$$$p^{32} T^{48} +$$$$45\!\cdots\!96$$$$p^{34} T^{49} +$$$$54\!\cdots\!16$$$$p^{36} T^{50} +$$$$25\!\cdots\!84$$$$p^{38} T^{51} -$$$$46\!\cdots\!44$$$$p^{40} T^{52} - 14505371407282045524 p^{42} T^{53} - 353884911424936344 p^{44} T^{54} - 9445294864601736 p^{46} T^{55} - 179847461096628 p^{48} T^{56} - 713393385972 p^{50} T^{57} + 89208581328 p^{52} T^{58} + 3910376532 p^{54} T^{59} + 102499780 p^{56} T^{60} + 1929672 p^{58} T^{61} + 25992 p^{60} T^{62} + 228 p^{62} T^{63} + p^{64} T^{64}$$
53 $$1 - 48 T + 1152 T^{2} - 26464 T^{3} + 12159320 T^{4} - 392146736 T^{5} + 5165678336 T^{6} + 489888623792 T^{7} + 81656780093668 T^{8} - 3756641219420384 T^{9} + 87364150697476736 T^{10} + 1265392886696245552 T^{11} +$$$$96\!\cdots\!04$$$$T^{12} -$$$$25\!\cdots\!56$$$$T^{13} +$$$$17\!\cdots\!40$$$$T^{14} -$$$$11\!\cdots\!20$$$$T^{15} +$$$$11\!\cdots\!26$$$$T^{16} -$$$$33\!\cdots\!00$$$$T^{17} +$$$$70\!\cdots\!64$$$$T^{18} -$$$$61\!\cdots\!20$$$$T^{19} +$$$$13\!\cdots\!12$$$$T^{20} -$$$$35\!\cdots\!64$$$$T^{21} +$$$$19\!\cdots\!88$$$$p T^{22} -$$$$29\!\cdots\!68$$$$T^{23} +$$$$97\!\cdots\!76$$$$T^{24} -$$$$22\!\cdots\!96$$$$T^{25} +$$$$16\!\cdots\!40$$$$T^{26} -$$$$38\!\cdots\!68$$$$T^{27} +$$$$85\!\cdots\!64$$$$T^{28} -$$$$19\!\cdots\!84$$$$T^{29} +$$$$60\!\cdots\!76$$$$T^{30} -$$$$30\!\cdots\!92$$$$T^{31} +$$$$65\!\cdots\!63$$$$T^{32} -$$$$30\!\cdots\!92$$$$p^{2} T^{33} +$$$$60\!\cdots\!76$$$$p^{4} T^{34} -$$$$19\!\cdots\!84$$$$p^{6} T^{35} +$$$$85\!\cdots\!64$$$$p^{8} T^{36} -$$$$38\!\cdots\!68$$$$p^{10} T^{37} +$$$$16\!\cdots\!40$$$$p^{12} T^{38} -$$$$22\!\cdots\!96$$$$p^{14} T^{39} +$$$$97\!\cdots\!76$$$$p^{16} T^{40} -$$$$29\!\cdots\!68$$$$p^{18} T^{41} +$$$$19\!\cdots\!88$$$$p^{21} T^{42} -$$$$35\!\cdots\!64$$$$p^{22} T^{43} +$$$$13\!\cdots\!12$$$$p^{24} T^{44} -$$$$61\!\cdots\!20$$$$p^{26} T^{45} +$$$$70\!\cdots\!64$$$$p^{28} T^{46} -$$$$33\!\cdots\!00$$$$p^{30} T^{47} +$$$$11\!\cdots\!26$$$$p^{32} T^{48} -$$$$11\!\cdots\!20$$$$p^{34} T^{49} +$$$$17\!\cdots\!40$$$$p^{36} T^{50} -$$$$25\!\cdots\!56$$$$p^{38} T^{51} +$$$$96\!\cdots\!04$$$$p^{40} T^{52} + 1265392886696245552 p^{42} T^{53} + 87364150697476736 p^{44} T^{54} - 3756641219420384 p^{46} T^{55} + 81656780093668 p^{48} T^{56} + 489888623792 p^{50} T^{57} + 5165678336 p^{52} T^{58} - 392146736 p^{54} T^{59} + 12159320 p^{56} T^{60} - 26464 p^{58} T^{61} + 1152 p^{60} T^{62} - 48 p^{62} T^{63} + p^{64} T^{64}$$
59 $$1 + 32408 T^{2} + 549062548 T^{4} + 6456694139464 T^{6} + 58949840848204124 T^{8} +$$$$44\!\cdots\!92$$$$T^{10} +$$$$28\!\cdots\!60$$$$T^{12} +$$$$16\!\cdots\!08$$$$T^{14} +$$$$79\!\cdots\!46$$$$T^{16} +$$$$35\!\cdots\!36$$$$T^{18} +$$$$14\!\cdots\!88$$$$T^{20} +$$$$51\!\cdots\!60$$$$T^{22} +$$$$16\!\cdots\!72$$$$T^{24} +$$$$50\!\cdots\!64$$$$T^{26} +$$$$14\!\cdots\!24$$$$T^{28} +$$$$38\!\cdots\!84$$$$T^{30} +$$$$12\!\cdots\!91$$$$T^{32} +$$$$38\!\cdots\!84$$$$p^{4} T^{34} +$$$$14\!\cdots\!24$$$$p^{8} T^{36} +$$$$50\!\cdots\!64$$$$p^{12} T^{38} +$$$$16\!\cdots\!72$$$$p^{16} T^{40} +$$$$51\!\cdots\!60$$$$p^{20} T^{42} +$$$$14\!\cdots\!88$$$$p^{24} T^{44} +$$$$35\!\cdots\!36$$$$p^{28} T^{46} +$$$$79\!\cdots\!46$$$$p^{32} T^{48} +$$$$16\!\cdots\!08$$$$p^{36} T^{50} +$$$$28\!\cdots\!60$$$$p^{40} T^{52} +$$$$44\!\cdots\!92$$$$p^{44} T^{54} + 58949840848204124 p^{48} T^{56} + 6456694139464 p^{52} T^{58} + 549062548 p^{56} T^{60} + 32408 p^{60} T^{62} + p^{64} T^{64}$$
61 $$( 1 - 108 T + 3344 T^{2} - 261456 T^{3} + 7251308 T^{4} + 1036857120 T^{5} + 18687170776 T^{6} - 210406697556 T^{7} - 304817639593378 T^{8} + 6951122192324784 T^{9} - 1315812464339416844 T^{10} + 76473784330953800400 T^{11} +$$$$22\!\cdots\!00$$$$T^{12} -$$$$85\!\cdots\!12$$$$T^{13} +$$$$45\!\cdots\!52$$$$T^{14} -$$$$38\!\cdots\!20$$$$T^{15} -$$$$43\!\cdots\!33$$$$T^{16} -$$$$38\!\cdots\!20$$$$p^{2} T^{17} +$$$$45\!\cdots\!52$$$$p^{4} T^{18} -$$$$85\!\cdots\!12$$$$p^{6} T^{19} +$$$$22\!\cdots\!00$$$$p^{8} T^{20} + 76473784330953800400 p^{10} T^{21} - 1315812464339416844 p^{12} T^{22} + 6951122192324784 p^{14} T^{23} - 304817639593378 p^{16} T^{24} - 210406697556 p^{18} T^{25} + 18687170776 p^{20} T^{26} + 1036857120 p^{22} T^{27} + 7251308 p^{24} T^{28} - 261456 p^{26} T^{29} + 3344 p^{28} T^{30} - 108 p^{30} T^{31} + p^{32} T^{32} )^{2}$$
67 $$1 + 3.85e5T^{3} + 3.38e7T^{4} + 2.72e9T^{5} + 7.41e10T^{6} + 2.09e13T^{7} + 1.53e15T^{8} + 8.23e16T^{9} + 9.24e18T^{10} + 6.75e20T^{11} + 6.56e22T^{12} + 3.13e24T^{13} + 2.76e26T^{14} + 2.47e28T^{15} + 1.70e30T^{16} + 1.24e32T^{17} + 7.20e33T^{18} + 6.58e35T^{19} + 4.22e37T^{20} + 2.93e39T^{21} + 1.96e41T^{22} + 1.40e43T^{23} + 1.10e45T^{24} + 5.83e46T^{25} + 4.40e48T^{26} + 2.81e50T^{27} + 2.28e52T^{28} + 1.40e54T^{29} + 7.97e55T^{30} + 6.60e57T^{31}+O(T^{32})$$
71 $$1 - 368T + 1.46e5T^{2} - 3.84e7T^{3} + 9.57e9T^{4} - 2.00e12T^{5} + 3.90e14T^{6} - 6.90e16T^{7} + 1.14e19T^{8} - 1.77e21T^{9} + 2.60e23T^{10} - 3.62e25T^{11} + 4.80e27T^{12} - 6.09e29T^{13} + 7.41e31T^{14} - 8.68e33T^{15} + 9.79e35T^{16} - 1.06e38T^{17} + 1.12e40T^{18} - 1.14e42T^{19} + 1.13e44T^{20} - 1.09e46T^{21} + 1.01e48T^{22} - 9.23e49T^{23} + 8.15e51T^{24} - 7.00e53T^{25} + 5.86e55T^{26} - 4.78e57T^{27} + 3.80e59T^{28} - 2.94e61T^{29} + 2.22e63T^{30} - 1.64e65T^{31}+O(T^{32})$$
73 $$1 - 52T + 1.35e3T^{2} + 1.65e6T^{3} - 8.01e7T^{4} + 4.24e9T^{5} + 1.26e12T^{6} - 6.27e13T^{7} + 5.24e15T^{8} + 5.66e17T^{9} - 3.04e19T^{10} + 3.88e21T^{11} + 1.63e23T^{12} - 1.18e25T^{13} + 2.13e27T^{14} + 4.64e28T^{15} - 5.57e30T^{16} + 1.02e33T^{17} + 2.21e34T^{18} - 2.88e36T^{19} + 4.91e38T^{20} + 9.52e39T^{21} - 1.12e42T^{22} + 2.28e44T^{23} + 1.78e45T^{24} - 2.42e47T^{25} + 9.83e49T^{26} - 4.35e50T^{27} - 1.90e52T^{28} + 3.90e55T^{29} - 2.75e56T^{30}+O(T^{31})$$
79 $$1 + 7.31e4T^{2} + 2.74e9T^{4} + 7.02e13T^{6} + 1.38e18T^{8} + 2.25e22T^{10} + 3.12e26T^{12} + 3.81e30T^{14} + 4.19e34T^{16} + 4.20e38T^{18} + 3.88e42T^{20} + 3.34e46T^{22} + 2.68e50T^{24} + 2.03e54T^{26} + 1.45e58T^{28} + 9.85e61T^{30}+O(T^{31})$$
83 $$1 + 736T + 2.70e5T^{2} + 6.83e7T^{3} + 1.37e10T^{4} + 2.41e12T^{5} + 3.80e14T^{6} + 5.48e16T^{7} + 7.29e18T^{8} + 9.10e20T^{9} + 1.07e23T^{10} + 1.20e25T^{11} + 1.30e27T^{12} + 1.36e29T^{13} + 1.37e31T^{14} + 1.36e33T^{15} + 1.33e35T^{16} + 1.27e37T^{17} + 1.21e39T^{18} + 1.14e41T^{19} + 1.05e43T^{20} + 9.74e44T^{21} + 8.85e46T^{22} + 7.97e48T^{23} + 7.11e50T^{24} + 6.29e52T^{25} + 5.52e54T^{26} + 4.82e56T^{27} + 4.17e58T^{28} + 3.58e60T^{29} + 3.05e62T^{30}+O(T^{31})$$
89 $$1 + 1.07e5T^{2} + 6.06e9T^{4} + 2.34e14T^{6} + 7.03e18T^{8} + 1.72e23T^{10} + 3.61e27T^{12} + 6.58e31T^{14} + 1.06e36T^{16} + 1.55e40T^{18} + 2.05e44T^{20} + 2.48e48T^{22} + 2.76e52T^{24} + 2.84e56T^{26} + 2.70e60T^{28}+O(T^{30})$$
97 $$1 + 408T + 8.32e4T^{2} + 1.62e7T^{3} + 3.12e9T^{4} + 4.86e11T^{5} + 7.02e13T^{6} + 1.03e16T^{7} + 1.39e18T^{8} + 1.80e20T^{9} + 2.36e22T^{10} + 3.00e24T^{11} + 3.69e26T^{12} + 4.53e28T^{13} + 5.46e30T^{14} + 6.43e32T^{15} + 7.47e34T^{16} + 8.55e36T^{17} + 9.67e38T^{18} + 1.07e41T^{19} + 1.18e43T^{20} + 1.29e45T^{21} + 1.40e47T^{22} + 1.49e49T^{23} + 1.57e51T^{24} + 1.64e53T^{25} + 1.69e55T^{26} + 1.73e57T^{27} + 1.75e59T^{28} + 1.76e61T^{29}+O(T^{30})$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{64} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}