# Properties

 Degree 68 Conductor $2^{68} \cdot 3^{34} \cdot 67^{34}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 34·3-s + 4-s − 4·7-s + 2·8-s + 595·9-s − 34·12-s + 16-s + 136·21-s − 68·24-s + 68·25-s − 7.14e3·27-s − 4·28-s − 16·29-s + 4·31-s + 2·32-s + 595·36-s + 12·37-s + 4·43-s − 34·48-s − 88·49-s − 8·56-s − 2.38e3·63-s + 7·64-s + 18·67-s + 1.19e3·72-s + 12·73-s − 2.31e3·75-s + ⋯
 L(s)  = 1 − 19.6·3-s + 1/2·4-s − 1.51·7-s + 0.707·8-s + 198.·9-s − 9.81·12-s + 1/4·16-s + 29.6·21-s − 13.8·24-s + 68/5·25-s − 1.37e3·27-s − 0.755·28-s − 2.97·29-s + 0.718·31-s + 0.353·32-s + 99.1·36-s + 1.97·37-s + 0.609·43-s − 4.90·48-s − 12.5·49-s − 1.06·56-s − 299.·63-s + 7/8·64-s + 2.19·67-s + 140.·72-s + 1.40·73-s − 266.·75-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{68} \cdot 3^{34} \cdot 67^{34}\right)^{s/2} \, \Gamma_{\C}(s)^{34} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{68} \cdot 3^{34} \cdot 67^{34}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{34} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 $$d$$ = $$68$$ $$N$$ = $$2^{68} \cdot 3^{34} \cdot 67^{34}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : induced by $\chi_{804} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(68,\ 2^{68} \cdot 3^{34} \cdot 67^{34} ,\ ( \ : [1/2]^{34} ),\ 1 )$ $L(1)$ $\approx$ $0.00828701$ $L(\frac12)$ $\approx$ $0.00828701$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;67\}$,$$F_p(T)$$ is a polynomial of degree 68. If $p \in \{2,\;3,\;67\}$, then $F_p(T)$ is a polynomial of degree at most 67.
$p$$F_p(T)$
bad2 $$1 - T^{2} - p T^{3} + p T^{5} - p T^{6} + p T^{7} + 3 T^{8} + 7 p T^{9} - T^{10} - 7 p^{3} T^{11} + 3 p^{4} T^{12} + 9 p^{3} T^{13} - p^{3} T^{14} - p^{6} T^{15} - 19 p^{3} T^{16} + p^{8} T^{17} - 19 p^{4} T^{18} - p^{8} T^{19} - p^{6} T^{20} + 9 p^{7} T^{21} + 3 p^{9} T^{22} - 7 p^{9} T^{23} - p^{7} T^{24} + 7 p^{9} T^{25} + 3 p^{9} T^{26} + p^{11} T^{27} - p^{12} T^{28} + p^{13} T^{29} - p^{15} T^{31} - p^{15} T^{32} + p^{17} T^{34}$$
3 $$( 1 + T )^{34}$$
67 $$1 - 18 T + 263 T^{2} - 2536 T^{3} + 15916 T^{4} + 21888 T^{5} - 1884372 T^{6} + 29838792 T^{7} - 281899248 T^{8} + 1676593384 T^{9} - 20990344 p T^{10} - 110944885160 T^{11} + 1642510138316 T^{12} - 13488669025792 T^{13} + 63391164577644 T^{14} + 2352888210968 p T^{15} - 6009516040445238 T^{16} + 66140757757941972 T^{17} - 6009516040445238 p T^{18} + 2352888210968 p^{3} T^{19} + 63391164577644 p^{3} T^{20} - 13488669025792 p^{4} T^{21} + 1642510138316 p^{5} T^{22} - 110944885160 p^{6} T^{23} - 20990344 p^{8} T^{24} + 1676593384 p^{8} T^{25} - 281899248 p^{9} T^{26} + 29838792 p^{10} T^{27} - 1884372 p^{11} T^{28} + 21888 p^{12} T^{29} + 15916 p^{13} T^{30} - 2536 p^{14} T^{31} + 263 p^{15} T^{32} - 18 p^{16} T^{33} + p^{17} T^{34}$$
good5 $$1 - 68 T^{2} + 2334 T^{4} - 53816 T^{6} + 935973 T^{8} - 13067332 T^{10} + 152110281 T^{12} - 302600004 p T^{14} + 13067728266 T^{16} - 98978994384 T^{18} + 660564070641 T^{20} - 3882218964872 T^{22} + 3990358248806 p T^{24} - 88176845698324 T^{26} + 64800202897848 p T^{28} - 925781797724652 T^{30} + 1839080435936258 T^{32} - 3866465311500328 T^{34} + 1839080435936258 p^{2} T^{36} - 925781797724652 p^{4} T^{38} + 64800202897848 p^{7} T^{40} - 88176845698324 p^{8} T^{42} + 3990358248806 p^{11} T^{44} - 3882218964872 p^{12} T^{46} + 660564070641 p^{14} T^{48} - 98978994384 p^{16} T^{50} + 13067728266 p^{18} T^{52} - 302600004 p^{21} T^{54} + 152110281 p^{22} T^{56} - 13067332 p^{24} T^{58} + 935973 p^{26} T^{60} - 53816 p^{28} T^{62} + 2334 p^{30} T^{64} - 68 p^{32} T^{66} + p^{34} T^{68}$$
7 $$( 1 + 2 T + 50 T^{2} + 104 T^{3} + 1219 T^{4} + 2696 T^{5} + 2827 p T^{6} + 960 p^{2} T^{7} + 248162 T^{8} + 622978 T^{9} + 2631571 T^{10} + 6666872 T^{11} + 24800242 T^{12} + 8606460 p T^{13} + 4318624 p^{2} T^{14} + 479245040 T^{15} + 1637496394 T^{16} + 3482409216 T^{17} + 1637496394 p T^{18} + 479245040 p^{2} T^{19} + 4318624 p^{5} T^{20} + 8606460 p^{5} T^{21} + 24800242 p^{5} T^{22} + 6666872 p^{6} T^{23} + 2631571 p^{7} T^{24} + 622978 p^{8} T^{25} + 248162 p^{9} T^{26} + 960 p^{12} T^{27} + 2827 p^{12} T^{28} + 2696 p^{12} T^{29} + 1219 p^{13} T^{30} + 104 p^{14} T^{31} + 50 p^{15} T^{32} + 2 p^{16} T^{33} + p^{17} T^{34} )^{2}$$
11 $$( 1 + 75 T^{2} - 68 T^{3} + 2884 T^{4} - 5024 T^{5} + 77848 T^{6} - 17244 p T^{7} + 1676376 T^{8} - 4910496 T^{9} + 30512488 T^{10} - 98245228 T^{11} + 481443968 T^{12} - 1613456096 T^{13} + 6673382908 T^{14} - 22402600284 T^{15} + 82108547374 T^{16} - 266083758016 T^{17} + 82108547374 p T^{18} - 22402600284 p^{2} T^{19} + 6673382908 p^{3} T^{20} - 1613456096 p^{4} T^{21} + 481443968 p^{5} T^{22} - 98245228 p^{6} T^{23} + 30512488 p^{7} T^{24} - 4910496 p^{8} T^{25} + 1676376 p^{9} T^{26} - 17244 p^{11} T^{27} + 77848 p^{11} T^{28} - 5024 p^{12} T^{29} + 2884 p^{13} T^{30} - 68 p^{14} T^{31} + 75 p^{15} T^{32} + p^{17} T^{34} )^{2}$$
13 $$1 - 202 T^{2} + 20513 T^{4} - 1397384 T^{6} + 71905920 T^{8} - 2984288512 T^{10} + 104162435968 T^{12} - 3147802485304 T^{14} + 6472168930596 p T^{16} - 2021525422384504 T^{18} + 44203317017326148 T^{20} - 888132807415483304 T^{22} + 16517338911140513536 T^{24} -$$$$28\!\cdots\!68$$$$T^{26} +$$$$46\!\cdots\!20$$$$T^{28} -$$$$70\!\cdots\!40$$$$T^{30} +$$$$10\!\cdots\!74$$$$T^{32} -$$$$13\!\cdots\!00$$$$T^{34} +$$$$10\!\cdots\!74$$$$p^{2} T^{36} -$$$$70\!\cdots\!40$$$$p^{4} T^{38} +$$$$46\!\cdots\!20$$$$p^{6} T^{40} -$$$$28\!\cdots\!68$$$$p^{8} T^{42} + 16517338911140513536 p^{10} T^{44} - 888132807415483304 p^{12} T^{46} + 44203317017326148 p^{14} T^{48} - 2021525422384504 p^{16} T^{50} + 6472168930596 p^{19} T^{52} - 3147802485304 p^{20} T^{54} + 104162435968 p^{22} T^{56} - 2984288512 p^{24} T^{58} + 71905920 p^{26} T^{60} - 1397384 p^{28} T^{62} + 20513 p^{30} T^{64} - 202 p^{32} T^{66} + p^{34} T^{68}$$
17 $$( 1 + 154 T^{2} - 10 T^{3} + 12007 T^{4} - 2236 T^{5} + 633281 T^{6} - 199606 T^{7} + 25348959 T^{8} - 10732600 T^{9} + 817172997 T^{10} - 408252386 T^{11} + 21953994331 T^{12} - 11874224100 T^{13} + 501614298749 T^{14} - 274906109790 T^{15} + 9860145355985 T^{16} - 5169284406096 T^{17} + 9860145355985 p T^{18} - 274906109790 p^{2} T^{19} + 501614298749 p^{3} T^{20} - 11874224100 p^{4} T^{21} + 21953994331 p^{5} T^{22} - 408252386 p^{6} T^{23} + 817172997 p^{7} T^{24} - 10732600 p^{8} T^{25} + 25348959 p^{9} T^{26} - 199606 p^{10} T^{27} + 633281 p^{11} T^{28} - 2236 p^{12} T^{29} + 12007 p^{13} T^{30} - 10 p^{14} T^{31} + 154 p^{15} T^{32} + p^{17} T^{34} )^{2}$$
19 $$1 - 324 T^{2} + 53806 T^{4} - 6082390 T^{6} + 524625555 T^{8} - 36706106300 T^{10} + 2163415764445 T^{12} - 110167923141938 T^{14} + 4934906438372087 T^{16} - 197035530538406632 T^{18} + 7082329410926741589 T^{20} -$$$$23\!\cdots\!26$$$$T^{22} +$$$$68\!\cdots\!71$$$$T^{24} -$$$$18\!\cdots\!28$$$$T^{26} +$$$$47\!\cdots\!01$$$$T^{28} -$$$$57\!\cdots\!34$$$$p T^{30} +$$$$23\!\cdots\!21$$$$T^{32} -$$$$45\!\cdots\!52$$$$T^{34} +$$$$23\!\cdots\!21$$$$p^{2} T^{36} -$$$$57\!\cdots\!34$$$$p^{5} T^{38} +$$$$47\!\cdots\!01$$$$p^{6} T^{40} -$$$$18\!\cdots\!28$$$$p^{8} T^{42} +$$$$68\!\cdots\!71$$$$p^{10} T^{44} -$$$$23\!\cdots\!26$$$$p^{12} T^{46} + 7082329410926741589 p^{14} T^{48} - 197035530538406632 p^{16} T^{50} + 4934906438372087 p^{18} T^{52} - 110167923141938 p^{20} T^{54} + 2163415764445 p^{22} T^{56} - 36706106300 p^{24} T^{58} + 524625555 p^{26} T^{60} - 6082390 p^{28} T^{62} + 53806 p^{30} T^{64} - 324 p^{32} T^{66} + p^{34} T^{68}$$
23 $$1 - 390 T^{2} + 77399 T^{4} - 452078 p T^{6} + 1061535793 T^{8} - 87680564496 T^{10} + 6092118022573 T^{12} - 365572082566210 T^{14} + 19305682322219015 T^{16} - 909807503167401174 T^{18} + 38667741337071246813 T^{20} -$$$$14\!\cdots\!96$$$$T^{22} +$$$$52\!\cdots\!10$$$$T^{24} -$$$$17\!\cdots\!68$$$$T^{26} +$$$$51\!\cdots\!70$$$$T^{28} -$$$$14\!\cdots\!44$$$$T^{30} +$$$$36\!\cdots\!18$$$$T^{32} -$$$$87\!\cdots\!56$$$$T^{34} +$$$$36\!\cdots\!18$$$$p^{2} T^{36} -$$$$14\!\cdots\!44$$$$p^{4} T^{38} +$$$$51\!\cdots\!70$$$$p^{6} T^{40} -$$$$17\!\cdots\!68$$$$p^{8} T^{42} +$$$$52\!\cdots\!10$$$$p^{10} T^{44} -$$$$14\!\cdots\!96$$$$p^{12} T^{46} + 38667741337071246813 p^{14} T^{48} - 909807503167401174 p^{16} T^{50} + 19305682322219015 p^{18} T^{52} - 365572082566210 p^{20} T^{54} + 6092118022573 p^{22} T^{56} - 87680564496 p^{24} T^{58} + 1061535793 p^{26} T^{60} - 452078 p^{29} T^{62} + 77399 p^{30} T^{64} - 390 p^{32} T^{66} + p^{34} T^{68}$$
29 $$( 1 + 8 T + 280 T^{2} + 1986 T^{3} + 38495 T^{4} + 249884 T^{5} + 120677 p T^{6} + 21143382 T^{7} + 237233865 T^{8} + 1346718616 T^{9} + 12796764223 T^{10} + 68628753698 T^{11} + 572314997735 T^{12} + 2906023079268 T^{13} + 21820589877857 T^{14} + 104770719965990 T^{15} + 722244133029203 T^{16} + 3261908176831296 T^{17} + 722244133029203 p T^{18} + 104770719965990 p^{2} T^{19} + 21820589877857 p^{3} T^{20} + 2906023079268 p^{4} T^{21} + 572314997735 p^{5} T^{22} + 68628753698 p^{6} T^{23} + 12796764223 p^{7} T^{24} + 1346718616 p^{8} T^{25} + 237233865 p^{9} T^{26} + 21143382 p^{10} T^{27} + 120677 p^{12} T^{28} + 249884 p^{12} T^{29} + 38495 p^{13} T^{30} + 1986 p^{14} T^{31} + 280 p^{15} T^{32} + 8 p^{16} T^{33} + p^{17} T^{34} )^{2}$$
31 $$( 1 - 2 T + 274 T^{2} - 438 T^{3} + 38283 T^{4} - 49456 T^{5} + 3629113 T^{6} - 3882350 T^{7} + 261988246 T^{8} - 239600890 T^{9} + 15286592715 T^{10} - 12308004016 T^{11} + 745422068666 T^{12} - 539717985316 T^{13} + 30972070210148 T^{14} - 20475487784252 T^{15} + 35764641776066 p T^{16} - 678116696787776 T^{17} + 35764641776066 p^{2} T^{18} - 20475487784252 p^{2} T^{19} + 30972070210148 p^{3} T^{20} - 539717985316 p^{4} T^{21} + 745422068666 p^{5} T^{22} - 12308004016 p^{6} T^{23} + 15286592715 p^{7} T^{24} - 239600890 p^{8} T^{25} + 261988246 p^{9} T^{26} - 3882350 p^{10} T^{27} + 3629113 p^{11} T^{28} - 49456 p^{12} T^{29} + 38283 p^{13} T^{30} - 438 p^{14} T^{31} + 274 p^{15} T^{32} - 2 p^{16} T^{33} + p^{17} T^{34} )^{2}$$
37 $$( 1 - 6 T + 291 T^{2} - 1114 T^{3} + 39201 T^{4} - 94944 T^{5} + 3531997 T^{6} - 5513746 T^{7} + 247356363 T^{8} - 263578318 T^{9} + 14354144765 T^{10} - 12006343760 T^{11} + 722409688070 T^{12} - 587638275932 T^{13} + 32491496385510 T^{14} - 27826635034132 T^{15} + 1320360130234934 T^{16} - 1125433059204576 T^{17} + 1320360130234934 p T^{18} - 27826635034132 p^{2} T^{19} + 32491496385510 p^{3} T^{20} - 587638275932 p^{4} T^{21} + 722409688070 p^{5} T^{22} - 12006343760 p^{6} T^{23} + 14354144765 p^{7} T^{24} - 263578318 p^{8} T^{25} + 247356363 p^{9} T^{26} - 5513746 p^{10} T^{27} + 3531997 p^{11} T^{28} - 94944 p^{12} T^{29} + 39201 p^{13} T^{30} - 1114 p^{14} T^{31} + 291 p^{15} T^{32} - 6 p^{16} T^{33} + p^{17} T^{34} )^{2}$$
41 $$1 - 800 T^{2} + 318522 T^{4} - 84180208 T^{6} + 16613975493 T^{8} - 2611646101660 T^{10} + 340551337922989 T^{12} - 37880873391773140 T^{14} + 3668463048471344246 T^{16} -$$$$31\!\cdots\!76$$$$T^{18} +$$$$58\!\cdots\!57$$$$p T^{20} -$$$$16\!\cdots\!68$$$$T^{22} +$$$$10\!\cdots\!74$$$$T^{24} -$$$$60\!\cdots\!88$$$$T^{26} +$$$$32\!\cdots\!56$$$$T^{28} -$$$$16\!\cdots\!04$$$$T^{30} +$$$$73\!\cdots\!46$$$$T^{32} -$$$$31\!\cdots\!52$$$$T^{34} +$$$$73\!\cdots\!46$$$$p^{2} T^{36} -$$$$16\!\cdots\!04$$$$p^{4} T^{38} +$$$$32\!\cdots\!56$$$$p^{6} T^{40} -$$$$60\!\cdots\!88$$$$p^{8} T^{42} +$$$$10\!\cdots\!74$$$$p^{10} T^{44} -$$$$16\!\cdots\!68$$$$p^{12} T^{46} +$$$$58\!\cdots\!57$$$$p^{15} T^{48} -$$$$31\!\cdots\!76$$$$p^{16} T^{50} + 3668463048471344246 p^{18} T^{52} - 37880873391773140 p^{20} T^{54} + 340551337922989 p^{22} T^{56} - 2611646101660 p^{24} T^{58} + 16613975493 p^{26} T^{60} - 84180208 p^{28} T^{62} + 318522 p^{30} T^{64} - 800 p^{32} T^{66} + p^{34} T^{68}$$
43 $$( 1 - 2 T + 416 T^{2} - 758 T^{3} + 84891 T^{4} - 141632 T^{5} + 11379069 T^{6} - 17778238 T^{7} + 1133402672 T^{8} - 1713556938 T^{9} + 89941328439 T^{10} - 135325872864 T^{11} + 5940357945610 T^{12} - 8972862446772 T^{13} + 335639372365200 T^{14} - 501267558316604 T^{15} + 16484044351581038 T^{16} - 23531265769512896 T^{17} + 16484044351581038 p T^{18} - 501267558316604 p^{2} T^{19} + 335639372365200 p^{3} T^{20} - 8972862446772 p^{4} T^{21} + 5940357945610 p^{5} T^{22} - 135325872864 p^{6} T^{23} + 89941328439 p^{7} T^{24} - 1713556938 p^{8} T^{25} + 1133402672 p^{9} T^{26} - 17778238 p^{10} T^{27} + 11379069 p^{11} T^{28} - 141632 p^{12} T^{29} + 84891 p^{13} T^{30} - 758 p^{14} T^{31} + 416 p^{15} T^{32} - 2 p^{16} T^{33} + p^{17} T^{34} )^{2}$$
47 $$1 - 786 T^{2} + 305642 T^{4} - 78650116 T^{6} + 15125906839 T^{8} - 2328618539216 T^{10} + 300069017373105 T^{12} - 33392461272792812 T^{14} + 3282424429226047787 T^{16} -$$$$28\!\cdots\!56$$$$T^{18} +$$$$23\!\cdots\!05$$$$T^{20} -$$$$17\!\cdots\!12$$$$T^{22} +$$$$11\!\cdots\!71$$$$T^{24} -$$$$73\!\cdots\!04$$$$T^{26} +$$$$43\!\cdots\!69$$$$T^{28} -$$$$24\!\cdots\!64$$$$T^{30} +$$$$12\!\cdots\!77$$$$T^{32} -$$$$60\!\cdots\!48$$$$T^{34} +$$$$12\!\cdots\!77$$$$p^{2} T^{36} -$$$$24\!\cdots\!64$$$$p^{4} T^{38} +$$$$43\!\cdots\!69$$$$p^{6} T^{40} -$$$$73\!\cdots\!04$$$$p^{8} T^{42} +$$$$11\!\cdots\!71$$$$p^{10} T^{44} -$$$$17\!\cdots\!12$$$$p^{12} T^{46} +$$$$23\!\cdots\!05$$$$p^{14} T^{48} -$$$$28\!\cdots\!56$$$$p^{16} T^{50} + 3282424429226047787 p^{18} T^{52} - 33392461272792812 p^{20} T^{54} + 300069017373105 p^{22} T^{56} - 2328618539216 p^{24} T^{58} + 15125906839 p^{26} T^{60} - 78650116 p^{28} T^{62} + 305642 p^{30} T^{64} - 786 p^{32} T^{66} + p^{34} T^{68}$$
53 $$1 - 1088 T^{2} + 593138 T^{4} - 215770592 T^{6} + 58848845469 T^{8} - 12818858766636 T^{10} + 2319881866245933 T^{12} - 358273125326199780 T^{14} + 48130466985201250542 T^{16} -$$$$57\!\cdots\!24$$$$T^{18} +$$$$60\!\cdots\!65$$$$T^{20} -$$$$10\!\cdots\!84$$$$p T^{22} +$$$$49\!\cdots\!38$$$$T^{24} -$$$$38\!\cdots\!28$$$$T^{26} +$$$$27\!\cdots\!32$$$$T^{28} -$$$$18\!\cdots\!08$$$$T^{30} +$$$$10\!\cdots\!98$$$$T^{32} -$$$$60\!\cdots\!84$$$$T^{34} +$$$$10\!\cdots\!98$$$$p^{2} T^{36} -$$$$18\!\cdots\!08$$$$p^{4} T^{38} +$$$$27\!\cdots\!32$$$$p^{6} T^{40} -$$$$38\!\cdots\!28$$$$p^{8} T^{42} +$$$$49\!\cdots\!38$$$$p^{10} T^{44} -$$$$10\!\cdots\!84$$$$p^{13} T^{46} +$$$$60\!\cdots\!65$$$$p^{14} T^{48} -$$$$57\!\cdots\!24$$$$p^{16} T^{50} + 48130466985201250542 p^{18} T^{52} - 358273125326199780 p^{20} T^{54} + 2319881866245933 p^{22} T^{56} - 12818858766636 p^{24} T^{58} + 58848845469 p^{26} T^{60} - 215770592 p^{28} T^{62} + 593138 p^{30} T^{64} - 1088 p^{32} T^{66} + p^{34} T^{68}$$
59 $$1 - 862 T^{2} + 368095 T^{4} - 103746158 T^{6} + 21709052577 T^{8} - 3601259631780 T^{10} + 494763257409217 T^{12} - 58212540248491374 T^{14} + 6035523432645758179 T^{16} -$$$$56\!\cdots\!02$$$$T^{18} +$$$$48\!\cdots\!13$$$$T^{20} -$$$$39\!\cdots\!16$$$$T^{22} +$$$$30\!\cdots\!14$$$$T^{24} -$$$$22\!\cdots\!88$$$$T^{26} +$$$$15\!\cdots\!62$$$$T^{28} -$$$$10\!\cdots\!16$$$$T^{30} +$$$$64\!\cdots\!50$$$$T^{32} -$$$$38\!\cdots\!84$$$$T^{34} +$$$$64\!\cdots\!50$$$$p^{2} T^{36} -$$$$10\!\cdots\!16$$$$p^{4} T^{38} +$$$$15\!\cdots\!62$$$$p^{6} T^{40} -$$$$22\!\cdots\!88$$$$p^{8} T^{42} +$$$$30\!\cdots\!14$$$$p^{10} T^{44} -$$$$39\!\cdots\!16$$$$p^{12} T^{46} +$$$$48\!\cdots\!13$$$$p^{14} T^{48} -$$$$56\!\cdots\!02$$$$p^{16} T^{50} + 6035523432645758179 p^{18} T^{52} - 58212540248491374 p^{20} T^{54} + 494763257409217 p^{22} T^{56} - 3601259631780 p^{24} T^{58} + 21709052577 p^{26} T^{60} - 103746158 p^{28} T^{62} + 368095 p^{30} T^{64} - 862 p^{32} T^{66} + p^{34} T^{68}$$
61 $$1 - 1306 T^{2} + 846889 T^{4} - 363644440 T^{6} + 116320882280 T^{8} - 29561600710304 T^{10} + 6215376728479224 T^{12} - 1111435431311189480 T^{14} +$$$$17\!\cdots\!16$$$$T^{16} -$$$$23\!\cdots\!00$$$$T^{18} +$$$$28\!\cdots\!40$$$$T^{20} -$$$$31\!\cdots\!68$$$$T^{22} +$$$$31\!\cdots\!12$$$$T^{24} -$$$$27\!\cdots\!16$$$$T^{26} +$$$$22\!\cdots\!12$$$$T^{28} -$$$$17\!\cdots\!92$$$$T^{30} +$$$$11\!\cdots\!62$$$$T^{32} -$$$$76\!\cdots\!20$$$$T^{34} +$$$$11\!\cdots\!62$$$$p^{2} T^{36} -$$$$17\!\cdots\!92$$$$p^{4} T^{38} +$$$$22\!\cdots\!12$$$$p^{6} T^{40} -$$$$27\!\cdots\!16$$$$p^{8} T^{42} +$$$$31\!\cdots\!12$$$$p^{10} T^{44} -$$$$31\!\cdots\!68$$$$p^{12} T^{46} +$$$$28\!\cdots\!40$$$$p^{14} T^{48} -$$$$23\!\cdots\!00$$$$p^{16} T^{50} +$$$$17\!\cdots\!16$$$$p^{18} T^{52} - 1111435431311189480 p^{20} T^{54} + 6215376728479224 p^{22} T^{56} - 29561600710304 p^{24} T^{58} + 116320882280 p^{26} T^{60} - 363644440 p^{28} T^{62} + 846889 p^{30} T^{64} - 1306 p^{32} T^{66} + p^{34} T^{68}$$
71 $$1 - 1056 T^{2} + 549229 T^{4} - 188159736 T^{6} + 47963906492 T^{8} - 9754150784304 T^{10} + 1658359742442852 T^{12} - 243960464630288360 T^{14} + 31880216274224415792 T^{16} -$$$$37\!\cdots\!40$$$$T^{18} +$$$$41\!\cdots\!80$$$$T^{20} -$$$$41\!\cdots\!72$$$$T^{22} +$$$$39\!\cdots\!04$$$$T^{24} -$$$$35\!\cdots\!08$$$$T^{26} +$$$$30\!\cdots\!48$$$$T^{28} -$$$$24\!\cdots\!88$$$$T^{30} +$$$$18\!\cdots\!70$$$$T^{32} -$$$$13\!\cdots\!96$$$$T^{34} +$$$$18\!\cdots\!70$$$$p^{2} T^{36} -$$$$24\!\cdots\!88$$$$p^{4} T^{38} +$$$$30\!\cdots\!48$$$$p^{6} T^{40} -$$$$35\!\cdots\!08$$$$p^{8} T^{42} +$$$$39\!\cdots\!04$$$$p^{10} T^{44} -$$$$41\!\cdots\!72$$$$p^{12} T^{46} +$$$$41\!\cdots\!80$$$$p^{14} T^{48} -$$$$37\!\cdots\!40$$$$p^{16} T^{50} + 31880216274224415792 p^{18} T^{52} - 243960464630288360 p^{20} T^{54} + 1658359742442852 p^{22} T^{56} - 9754150784304 p^{24} T^{58} + 47963906492 p^{26} T^{60} - 188159736 p^{28} T^{62} + 549229 p^{30} T^{64} - 1056 p^{32} T^{66} + p^{34} T^{68}$$
73 $$( 1 - 6 T + 515 T^{2} - 3328 T^{3} + 144541 T^{4} - 940110 T^{5} + 28581217 T^{6} - 180185960 T^{7} + 4391332191 T^{8} - 26455644678 T^{9} + 553385774101 T^{10} - 3166334384520 T^{11} + 59248572421426 T^{12} - 321034920001700 T^{13} + 5516399267831014 T^{14} - 28277250519852264 T^{15} + 453126281475821258 T^{16} - 2194400317569554068 T^{17} + 453126281475821258 p T^{18} - 28277250519852264 p^{2} T^{19} + 5516399267831014 p^{3} T^{20} - 321034920001700 p^{4} T^{21} + 59248572421426 p^{5} T^{22} - 3166334384520 p^{6} T^{23} + 553385774101 p^{7} T^{24} - 26455644678 p^{8} T^{25} + 4391332191 p^{9} T^{26} - 180185960 p^{10} T^{27} + 28581217 p^{11} T^{28} - 940110 p^{12} T^{29} + 144541 p^{13} T^{30} - 3328 p^{14} T^{31} + 515 p^{15} T^{32} - 6 p^{16} T^{33} + p^{17} T^{34} )^{2}$$
79 $$( 1 - 6 T + 865 T^{2} - 4056 T^{3} + 364802 T^{4} - 1343440 T^{5} + 100925602 T^{6} - 294801704 T^{7} + 20670590462 T^{8} - 48642581736 T^{9} + 3336713662750 T^{10} - 6461771515768 T^{11} + 440013510687962 T^{12} - 720657077267376 T^{13} + 48426256361249146 T^{14} - 69315757086640520 T^{15} + 4502866981735763504 T^{16} - 5837866981735029028 T^{17} + 4502866981735763504 p T^{18} - 69315757086640520 p^{2} T^{19} + 48426256361249146 p^{3} T^{20} - 720657077267376 p^{4} T^{21} + 440013510687962 p^{5} T^{22} - 6461771515768 p^{6} T^{23} + 3336713662750 p^{7} T^{24} - 48642581736 p^{8} T^{25} + 20670590462 p^{9} T^{26} - 294801704 p^{10} T^{27} + 100925602 p^{11} T^{28} - 1343440 p^{12} T^{29} + 364802 p^{13} T^{30} - 4056 p^{14} T^{31} + 865 p^{15} T^{32} - 6 p^{16} T^{33} + p^{17} T^{34} )^{2}$$
83 $$1 - 1292 T^{2} + 848290 T^{4} - 376704388 T^{6} + 127097108205 T^{8} - 34708595206900 T^{10} + 7983013214105213 T^{12} - 1589072191555694912 T^{14} +$$$$27\!\cdots\!38$$$$T^{16} -$$$$43\!\cdots\!24$$$$T^{18} +$$$$62\!\cdots\!01$$$$T^{20} -$$$$81\!\cdots\!72$$$$T^{22} +$$$$98\!\cdots\!82$$$$T^{24} -$$$$11\!\cdots\!68$$$$T^{26} +$$$$11\!\cdots\!84$$$$T^{28} -$$$$11\!\cdots\!36$$$$T^{30} +$$$$10\!\cdots\!30$$$$T^{32} -$$$$86\!\cdots\!56$$$$T^{34} +$$$$10\!\cdots\!30$$$$p^{2} T^{36} -$$$$11\!\cdots\!36$$$$p^{4} T^{38} +$$$$11\!\cdots\!84$$$$p^{6} T^{40} -$$$$11\!\cdots\!68$$$$p^{8} T^{42} +$$$$98\!\cdots\!82$$$$p^{10} T^{44} -$$$$81\!\cdots\!72$$$$p^{12} T^{46} +$$$$62\!\cdots\!01$$$$p^{14} T^{48} -$$$$43\!\cdots\!24$$$$p^{16} T^{50} +$$$$27\!\cdots\!38$$$$p^{18} T^{52} - 1589072191555694912 p^{20} T^{54} + 7983013214105213 p^{22} T^{56} - 34708595206900 p^{24} T^{58} + 127097108205 p^{26} T^{60} - 376704388 p^{28} T^{62} + 848290 p^{30} T^{64} - 1292 p^{32} T^{66} + p^{34} T^{68}$$
89 $$( 1 + 638 T^{2} - 1230 T^{3} + 210827 T^{4} - 839464 T^{5} + 48519761 T^{6} - 283155618 T^{7} + 8787644647 T^{8} - 64303304896 T^{9} + 1328098248333 T^{10} - 11104840293638 T^{11} + 172455439401115 T^{12} - 1544526529348152 T^{13} + 19580679895915225 T^{14} - 178000703040889418 T^{15} + 1964090161809709429 T^{16} - 17240520658784583360 T^{17} + 1964090161809709429 p T^{18} - 178000703040889418 p^{2} T^{19} + 19580679895915225 p^{3} T^{20} - 1544526529348152 p^{4} T^{21} + 172455439401115 p^{5} T^{22} - 11104840293638 p^{6} T^{23} + 1328098248333 p^{7} T^{24} - 64303304896 p^{8} T^{25} + 8787644647 p^{9} T^{26} - 283155618 p^{10} T^{27} + 48519761 p^{11} T^{28} - 839464 p^{12} T^{29} + 210827 p^{13} T^{30} - 1230 p^{14} T^{31} + 638 p^{15} T^{32} + p^{17} T^{34} )^{2}$$
97 $$1 - 1830 T^{2} + 1694021 T^{4} - 1054854944 T^{6} + 495970293140 T^{8} - 187442459787088 T^{10} + 59203185437163852 T^{12} - 16044756844465501184 T^{14} +$$$$38\!\cdots\!88$$$$T^{16} -$$$$82\!\cdots\!48$$$$p T^{18} +$$$$15\!\cdots\!44$$$$T^{20} -$$$$25\!\cdots\!48$$$$T^{22} +$$$$39\!\cdots\!80$$$$T^{24} -$$$$55\!\cdots\!76$$$$T^{26} +$$$$72\!\cdots\!76$$$$T^{28} -$$$$85\!\cdots\!52$$$$T^{30} +$$$$94\!\cdots\!86$$$$T^{32} -$$$$95\!\cdots\!44$$$$T^{34} +$$$$94\!\cdots\!86$$$$p^{2} T^{36} -$$$$85\!\cdots\!52$$$$p^{4} T^{38} +$$$$72\!\cdots\!76$$$$p^{6} T^{40} -$$$$55\!\cdots\!76$$$$p^{8} T^{42} +$$$$39\!\cdots\!80$$$$p^{10} T^{44} -$$$$25\!\cdots\!48$$$$p^{12} T^{46} +$$$$15\!\cdots\!44$$$$p^{14} T^{48} -$$$$82\!\cdots\!48$$$$p^{17} T^{50} +$$$$38\!\cdots\!88$$$$p^{18} T^{52} - 16044756844465501184 p^{20} T^{54} + 59203185437163852 p^{22} T^{56} - 187442459787088 p^{24} T^{58} + 495970293140 p^{26} T^{60} - 1054854944 p^{28} T^{62} + 1694021 p^{30} T^{64} - 1830 p^{32} T^{66} + p^{34} T^{68}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{68} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}