# Properties

 Label 32-1617e16-1.1-c3e16-0-2 Degree $32$ Conductor $2.185\times 10^{51}$ Sign $1$ Analytic cond. $4.71213\times 10^{31}$ Root an. cond. $9.76760$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $16$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4·2-s − 48·3-s − 20·4-s − 40·5-s − 192·6-s − 90·8-s + 1.22e3·9-s − 160·10-s + 176·11-s + 960·12-s − 104·13-s + 1.92e3·15-s + 135·16-s − 180·17-s + 4.89e3·18-s − 152·19-s + 800·20-s + 704·22-s + 4·23-s + 4.32e3·24-s + 94·25-s − 416·26-s − 2.20e4·27-s + 412·29-s + 7.68e3·30-s − 628·31-s + 708·32-s + ⋯
 L(s)  = 1 + 1.41·2-s − 9.23·3-s − 5/2·4-s − 3.57·5-s − 13.0·6-s − 3.97·8-s + 45.3·9-s − 5.05·10-s + 4.82·11-s + 23.0·12-s − 2.21·13-s + 33.0·15-s + 2.10·16-s − 2.56·17-s + 64.1·18-s − 1.83·19-s + 8.94·20-s + 6.82·22-s + 0.0362·23-s + 36.7·24-s + 0.751·25-s − 3.13·26-s − 157.·27-s + 2.63·29-s + 46.7·30-s − 3.63·31-s + 3.91·32-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{32} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{32} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$32$$ Conductor: $$3^{16} \cdot 7^{32} \cdot 11^{16}$$ Sign: $1$ Analytic conductor: $$4.71213\times 10^{31}$$ Root analytic conductor: $$9.76760$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$16$$ Selberg data: $$(32,\ 3^{16} \cdot 7^{32} \cdot 11^{16} ,\ ( \ : [3/2]^{16} ),\ 1 )$$

## Particular Values

 $$L(2)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$( 1 + p T )^{16}$$
7 $$1$$
11 $$( 1 - p T )^{16}$$
good2 $$1 - p^{2} T + 9 p^{2} T^{2} - 67 p T^{3} + 761 T^{4} - 663 p^{2} T^{5} + 11997 T^{6} - 19643 p T^{7} + 78699 p T^{8} - 121701 p^{2} T^{9} + 1783439 T^{10} - 327623 p^{4} T^{11} + 1122389 p^{4} T^{12} - 1570281 p^{5} T^{13} + 10214477 p^{4} T^{14} - 853231 p^{9} T^{15} + 5325245 p^{8} T^{16} - 853231 p^{12} T^{17} + 10214477 p^{10} T^{18} - 1570281 p^{14} T^{19} + 1122389 p^{16} T^{20} - 327623 p^{19} T^{21} + 1783439 p^{18} T^{22} - 121701 p^{23} T^{23} + 78699 p^{25} T^{24} - 19643 p^{28} T^{25} + 11997 p^{30} T^{26} - 663 p^{35} T^{27} + 761 p^{36} T^{28} - 67 p^{40} T^{29} + 9 p^{44} T^{30} - p^{47} T^{31} + p^{48} T^{32}$$
5 $$1 + 8 p T + 1506 T^{2} + 40108 T^{3} + 198522 p T^{4} + 20758872 T^{5} + 409277034 T^{6} + 7248970152 T^{7} + 122325311177 T^{8} + 1907513437252 T^{9} + 28563727485808 T^{10} + 401389638566252 T^{11} + 5443835695315814 T^{12} + 13977776900911504 p T^{13} + 173740728130868532 p T^{14} + 2053249282975640876 p T^{15} +$$$$11\!\cdots\!48$$$$T^{16} + 2053249282975640876 p^{4} T^{17} + 173740728130868532 p^{7} T^{18} + 13977776900911504 p^{10} T^{19} + 5443835695315814 p^{12} T^{20} + 401389638566252 p^{15} T^{21} + 28563727485808 p^{18} T^{22} + 1907513437252 p^{21} T^{23} + 122325311177 p^{24} T^{24} + 7248970152 p^{27} T^{25} + 409277034 p^{30} T^{26} + 20758872 p^{33} T^{27} + 198522 p^{37} T^{28} + 40108 p^{39} T^{29} + 1506 p^{42} T^{30} + 8 p^{46} T^{31} + p^{48} T^{32}$$
13 $$1 + 8 p T + 25652 T^{2} + 1943184 T^{3} + 279130654 T^{4} + 16124021432 T^{5} + 1781810145360 T^{6} + 79381828257816 T^{7} + 7795017371133297 T^{8} + 267294898729036520 T^{9} + 26384331011517156904 T^{10} +$$$$70\!\cdots\!20$$$$T^{11} +$$$$59\!\cdots\!42$$$$p T^{12} +$$$$17\!\cdots\!76$$$$T^{13} +$$$$20\!\cdots\!12$$$$T^{14} +$$$$40\!\cdots\!52$$$$T^{15} +$$$$47\!\cdots\!04$$$$T^{16} +$$$$40\!\cdots\!52$$$$p^{3} T^{17} +$$$$20\!\cdots\!12$$$$p^{6} T^{18} +$$$$17\!\cdots\!76$$$$p^{9} T^{19} +$$$$59\!\cdots\!42$$$$p^{13} T^{20} +$$$$70\!\cdots\!20$$$$p^{15} T^{21} + 26384331011517156904 p^{18} T^{22} + 267294898729036520 p^{21} T^{23} + 7795017371133297 p^{24} T^{24} + 79381828257816 p^{27} T^{25} + 1781810145360 p^{30} T^{26} + 16124021432 p^{33} T^{27} + 279130654 p^{36} T^{28} + 1943184 p^{39} T^{29} + 25652 p^{42} T^{30} + 8 p^{46} T^{31} + p^{48} T^{32}$$
17 $$1 + 180 T + 54198 T^{2} + 7957236 T^{3} + 1451362768 T^{4} + 181850061036 T^{5} + 25448994074426 T^{6} + 2795276612168940 T^{7} + 326768347236760204 T^{8} + 32004380790033019140 T^{9} +$$$$19\!\cdots\!02$$$$p T^{10} +$$$$28\!\cdots\!08$$$$T^{11} +$$$$26\!\cdots\!88$$$$T^{12} +$$$$20\!\cdots\!12$$$$T^{13} +$$$$16\!\cdots\!66$$$$T^{14} +$$$$12\!\cdots\!00$$$$T^{15} +$$$$53\!\cdots\!62$$$$p T^{16} +$$$$12\!\cdots\!00$$$$p^{3} T^{17} +$$$$16\!\cdots\!66$$$$p^{6} T^{18} +$$$$20\!\cdots\!12$$$$p^{9} T^{19} +$$$$26\!\cdots\!88$$$$p^{12} T^{20} +$$$$28\!\cdots\!08$$$$p^{15} T^{21} +$$$$19\!\cdots\!02$$$$p^{19} T^{22} + 32004380790033019140 p^{21} T^{23} + 326768347236760204 p^{24} T^{24} + 2795276612168940 p^{27} T^{25} + 25448994074426 p^{30} T^{26} + 181850061036 p^{33} T^{27} + 1451362768 p^{36} T^{28} + 7957236 p^{39} T^{29} + 54198 p^{42} T^{30} + 180 p^{45} T^{31} + p^{48} T^{32}$$
19 $$1 + 8 p T + 72362 T^{2} + 7737068 T^{3} + 2200706030 T^{4} + 164945433760 T^{5} + 38804955600054 T^{6} + 1798778642354344 T^{7} + 453186557872646361 T^{8} + 6241087305607852980 T^{9} +$$$$37\!\cdots\!04$$$$T^{10} -$$$$12\!\cdots\!84$$$$T^{11} +$$$$22\!\cdots\!74$$$$T^{12} -$$$$25\!\cdots\!24$$$$T^{13} +$$$$10\!\cdots\!40$$$$T^{14} -$$$$25\!\cdots\!88$$$$T^{15} +$$$$51\!\cdots\!72$$$$T^{16} -$$$$25\!\cdots\!88$$$$p^{3} T^{17} +$$$$10\!\cdots\!40$$$$p^{6} T^{18} -$$$$25\!\cdots\!24$$$$p^{9} T^{19} +$$$$22\!\cdots\!74$$$$p^{12} T^{20} -$$$$12\!\cdots\!84$$$$p^{15} T^{21} +$$$$37\!\cdots\!04$$$$p^{18} T^{22} + 6241087305607852980 p^{21} T^{23} + 453186557872646361 p^{24} T^{24} + 1798778642354344 p^{27} T^{25} + 38804955600054 p^{30} T^{26} + 164945433760 p^{33} T^{27} + 2200706030 p^{36} T^{28} + 7737068 p^{39} T^{29} + 72362 p^{42} T^{30} + 8 p^{46} T^{31} + p^{48} T^{32}$$
23 $$1 - 4 T + 92096 T^{2} + 851484 T^{3} + 181757852 p T^{4} + 96146813964 T^{5} + 126657388879536 T^{6} + 198373717336820 p T^{7} + 2933207092863655876 T^{8} +$$$$13\!\cdots\!08$$$$T^{9} +$$$$56\!\cdots\!68$$$$T^{10} +$$$$13\!\cdots\!20$$$$p T^{11} +$$$$92\!\cdots\!92$$$$T^{12} +$$$$55\!\cdots\!64$$$$T^{13} +$$$$13\!\cdots\!88$$$$T^{14} +$$$$81\!\cdots\!04$$$$T^{15} +$$$$17\!\cdots\!26$$$$T^{16} +$$$$81\!\cdots\!04$$$$p^{3} T^{17} +$$$$13\!\cdots\!88$$$$p^{6} T^{18} +$$$$55\!\cdots\!64$$$$p^{9} T^{19} +$$$$92\!\cdots\!92$$$$p^{12} T^{20} +$$$$13\!\cdots\!20$$$$p^{16} T^{21} +$$$$56\!\cdots\!68$$$$p^{18} T^{22} +$$$$13\!\cdots\!08$$$$p^{21} T^{23} + 2933207092863655876 p^{24} T^{24} + 198373717336820 p^{28} T^{25} + 126657388879536 p^{30} T^{26} + 96146813964 p^{33} T^{27} + 181757852 p^{37} T^{28} + 851484 p^{39} T^{29} + 92096 p^{42} T^{30} - 4 p^{45} T^{31} + p^{48} T^{32}$$
29 $$1 - 412 T + 300056 T^{2} - 86571584 T^{3} + 36823296078 T^{4} - 8095039495788 T^{5} + 2611046195789656 T^{6} - 454346770907438348 T^{7} +$$$$12\!\cdots\!97$$$$T^{8} -$$$$17\!\cdots\!72$$$$T^{9} +$$$$46\!\cdots\!80$$$$T^{10} -$$$$54\!\cdots\!48$$$$T^{11} +$$$$14\!\cdots\!06$$$$T^{12} -$$$$15\!\cdots\!56$$$$T^{13} +$$$$41\!\cdots\!20$$$$T^{14} -$$$$39\!\cdots\!88$$$$T^{15} +$$$$10\!\cdots\!68$$$$T^{16} -$$$$39\!\cdots\!88$$$$p^{3} T^{17} +$$$$41\!\cdots\!20$$$$p^{6} T^{18} -$$$$15\!\cdots\!56$$$$p^{9} T^{19} +$$$$14\!\cdots\!06$$$$p^{12} T^{20} -$$$$54\!\cdots\!48$$$$p^{15} T^{21} +$$$$46\!\cdots\!80$$$$p^{18} T^{22} -$$$$17\!\cdots\!72$$$$p^{21} T^{23} +$$$$12\!\cdots\!97$$$$p^{24} T^{24} - 454346770907438348 p^{27} T^{25} + 2611046195789656 p^{30} T^{26} - 8095039495788 p^{33} T^{27} + 36823296078 p^{36} T^{28} - 86571584 p^{39} T^{29} + 300056 p^{42} T^{30} - 412 p^{45} T^{31} + p^{48} T^{32}$$
31 $$1 + 628 T + 443802 T^{2} + 178346516 T^{3} + 74374585860 T^{4} + 22592241795284 T^{5} + 7086480563914142 T^{6} + 1769074867387298804 T^{7} +$$$$46\!\cdots\!96$$$$T^{8} +$$$$99\!\cdots\!40$$$$T^{9} +$$$$22\!\cdots\!34$$$$T^{10} +$$$$43\!\cdots\!96$$$$T^{11} +$$$$90\!\cdots\!56$$$$T^{12} +$$$$16\!\cdots\!68$$$$T^{13} +$$$$31\!\cdots\!06$$$$T^{14} +$$$$52\!\cdots\!96$$$$T^{15} +$$$$96\!\cdots\!06$$$$T^{16} +$$$$52\!\cdots\!96$$$$p^{3} T^{17} +$$$$31\!\cdots\!06$$$$p^{6} T^{18} +$$$$16\!\cdots\!68$$$$p^{9} T^{19} +$$$$90\!\cdots\!56$$$$p^{12} T^{20} +$$$$43\!\cdots\!96$$$$p^{15} T^{21} +$$$$22\!\cdots\!34$$$$p^{18} T^{22} +$$$$99\!\cdots\!40$$$$p^{21} T^{23} +$$$$46\!\cdots\!96$$$$p^{24} T^{24} + 1769074867387298804 p^{27} T^{25} + 7086480563914142 p^{30} T^{26} + 22592241795284 p^{33} T^{27} + 74374585860 p^{36} T^{28} + 178346516 p^{39} T^{29} + 443802 p^{42} T^{30} + 628 p^{45} T^{31} + p^{48} T^{32}$$
37 $$1 - 4 p T + 411080 T^{2} - 70256444 T^{3} + 83559740790 T^{4} - 16074136192876 T^{5} + 11346747767975180 T^{6} - 64294568607886452 p T^{7} +$$$$11\!\cdots\!21$$$$T^{8} -$$$$25\!\cdots\!32$$$$T^{9} +$$$$10\!\cdots\!00$$$$T^{10} -$$$$21\!\cdots\!64$$$$T^{11} +$$$$72\!\cdots\!74$$$$T^{12} -$$$$15\!\cdots\!48$$$$T^{13} +$$$$45\!\cdots\!16$$$$T^{14} -$$$$90\!\cdots\!40$$$$T^{15} +$$$$25\!\cdots\!64$$$$T^{16} -$$$$90\!\cdots\!40$$$$p^{3} T^{17} +$$$$45\!\cdots\!16$$$$p^{6} T^{18} -$$$$15\!\cdots\!48$$$$p^{9} T^{19} +$$$$72\!\cdots\!74$$$$p^{12} T^{20} -$$$$21\!\cdots\!64$$$$p^{15} T^{21} +$$$$10\!\cdots\!00$$$$p^{18} T^{22} -$$$$25\!\cdots\!32$$$$p^{21} T^{23} +$$$$11\!\cdots\!21$$$$p^{24} T^{24} - 64294568607886452 p^{28} T^{25} + 11346747767975180 p^{30} T^{26} - 16074136192876 p^{33} T^{27} + 83559740790 p^{36} T^{28} - 70256444 p^{39} T^{29} + 411080 p^{42} T^{30} - 4 p^{46} T^{31} + p^{48} T^{32}$$
41 $$1 + 596 T + 558130 T^{2} + 284481652 T^{3} + 160254133820 T^{4} + 68353164664380 T^{5} + 29609167020868838 T^{6} + 10810190321839250780 T^{7} +$$$$39\!\cdots\!84$$$$T^{8} +$$$$12\!\cdots\!76$$$$T^{9} +$$$$39\!\cdots\!78$$$$T^{10} +$$$$11\!\cdots\!56$$$$T^{11} +$$$$77\!\cdots\!16$$$$p T^{12} +$$$$84\!\cdots\!00$$$$T^{13} +$$$$22\!\cdots\!86$$$$T^{14} +$$$$57\!\cdots\!12$$$$T^{15} +$$$$15\!\cdots\!34$$$$T^{16} +$$$$57\!\cdots\!12$$$$p^{3} T^{17} +$$$$22\!\cdots\!86$$$$p^{6} T^{18} +$$$$84\!\cdots\!00$$$$p^{9} T^{19} +$$$$77\!\cdots\!16$$$$p^{13} T^{20} +$$$$11\!\cdots\!56$$$$p^{15} T^{21} +$$$$39\!\cdots\!78$$$$p^{18} T^{22} +$$$$12\!\cdots\!76$$$$p^{21} T^{23} +$$$$39\!\cdots\!84$$$$p^{24} T^{24} + 10810190321839250780 p^{27} T^{25} + 29609167020868838 p^{30} T^{26} + 68353164664380 p^{33} T^{27} + 160254133820 p^{36} T^{28} + 284481652 p^{39} T^{29} + 558130 p^{42} T^{30} + 596 p^{45} T^{31} + p^{48} T^{32}$$
43 $$1 + 260 T + 364520 T^{2} + 55410228 T^{3} + 52760767684 T^{4} + 3409970749748 T^{5} + 4384742136277704 T^{6} - 32727427717038300 T^{7} +$$$$26\!\cdots\!40$$$$T^{8} -$$$$12\!\cdots\!56$$$$T^{9} +$$$$13\!\cdots\!76$$$$T^{10} +$$$$81\!\cdots\!28$$$$T^{11} +$$$$95\!\cdots\!20$$$$T^{12} +$$$$36\!\cdots\!36$$$$T^{13} +$$$$88\!\cdots\!48$$$$T^{14} +$$$$46\!\cdots\!40$$$$T^{15} +$$$$73\!\cdots\!86$$$$T^{16} +$$$$46\!\cdots\!40$$$$p^{3} T^{17} +$$$$88\!\cdots\!48$$$$p^{6} T^{18} +$$$$36\!\cdots\!36$$$$p^{9} T^{19} +$$$$95\!\cdots\!20$$$$p^{12} T^{20} +$$$$81\!\cdots\!28$$$$p^{15} T^{21} +$$$$13\!\cdots\!76$$$$p^{18} T^{22} -$$$$12\!\cdots\!56$$$$p^{21} T^{23} +$$$$26\!\cdots\!40$$$$p^{24} T^{24} - 32727427717038300 p^{27} T^{25} + 4384742136277704 p^{30} T^{26} + 3409970749748 p^{33} T^{27} + 52760767684 p^{36} T^{28} + 55410228 p^{39} T^{29} + 364520 p^{42} T^{30} + 260 p^{45} T^{31} + p^{48} T^{32}$$
47 $$1 + 2220 T + 3250782 T^{2} + 3488867780 T^{3} + 3068601127966 T^{4} + 2279830005108700 T^{5} + 1480141931179634422 T^{6} +$$$$85\!\cdots\!12$$$$T^{7} +$$$$44\!\cdots\!21$$$$T^{8} +$$$$20\!\cdots\!60$$$$T^{9} +$$$$89\!\cdots\!80$$$$T^{10} +$$$$35\!\cdots\!08$$$$T^{11} +$$$$13\!\cdots\!54$$$$T^{12} +$$$$47\!\cdots\!56$$$$T^{13} +$$$$16\!\cdots\!48$$$$T^{14} +$$$$52\!\cdots\!28$$$$T^{15} +$$$$17\!\cdots\!12$$$$T^{16} +$$$$52\!\cdots\!28$$$$p^{3} T^{17} +$$$$16\!\cdots\!48$$$$p^{6} T^{18} +$$$$47\!\cdots\!56$$$$p^{9} T^{19} +$$$$13\!\cdots\!54$$$$p^{12} T^{20} +$$$$35\!\cdots\!08$$$$p^{15} T^{21} +$$$$89\!\cdots\!80$$$$p^{18} T^{22} +$$$$20\!\cdots\!60$$$$p^{21} T^{23} +$$$$44\!\cdots\!21$$$$p^{24} T^{24} +$$$$85\!\cdots\!12$$$$p^{27} T^{25} + 1480141931179634422 p^{30} T^{26} + 2279830005108700 p^{33} T^{27} + 3068601127966 p^{36} T^{28} + 3488867780 p^{39} T^{29} + 3250782 p^{42} T^{30} + 2220 p^{45} T^{31} + p^{48} T^{32}$$
53 $$1 - 168 T + 1584380 T^{2} - 377434888 T^{3} + 1226513279676 T^{4} - 366057928565944 T^{5} + 623474858770816212 T^{6} -$$$$21\!\cdots\!24$$$$T^{7} +$$$$23\!\cdots\!76$$$$T^{8} -$$$$85\!\cdots\!96$$$$T^{9} +$$$$69\!\cdots\!72$$$$T^{10} -$$$$25\!\cdots\!40$$$$T^{11} +$$$$16\!\cdots\!44$$$$T^{12} -$$$$58\!\cdots\!04$$$$T^{13} +$$$$33\!\cdots\!36$$$$T^{14} -$$$$10\!\cdots\!24$$$$T^{15} +$$$$54\!\cdots\!46$$$$T^{16} -$$$$10\!\cdots\!24$$$$p^{3} T^{17} +$$$$33\!\cdots\!36$$$$p^{6} T^{18} -$$$$58\!\cdots\!04$$$$p^{9} T^{19} +$$$$16\!\cdots\!44$$$$p^{12} T^{20} -$$$$25\!\cdots\!40$$$$p^{15} T^{21} +$$$$69\!\cdots\!72$$$$p^{18} T^{22} -$$$$85\!\cdots\!96$$$$p^{21} T^{23} +$$$$23\!\cdots\!76$$$$p^{24} T^{24} -$$$$21\!\cdots\!24$$$$p^{27} T^{25} + 623474858770816212 p^{30} T^{26} - 366057928565944 p^{33} T^{27} + 1226513279676 p^{36} T^{28} - 377434888 p^{39} T^{29} + 1584380 p^{42} T^{30} - 168 p^{45} T^{31} + p^{48} T^{32}$$
59 $$1 - 48 T + 1866728 T^{2} - 25998216 T^{3} + 1736544062022 T^{4} + 39208048973064 T^{5} + 1073950381186450240 T^{6} + 61873064288651078904 T^{7} +$$$$49\!\cdots\!61$$$$T^{8} +$$$$43\!\cdots\!72$$$$T^{9} +$$$$18\!\cdots\!52$$$$T^{10} +$$$$20\!\cdots\!80$$$$T^{11} +$$$$55\!\cdots\!22$$$$T^{12} +$$$$11\!\cdots\!24$$$$p T^{13} +$$$$14\!\cdots\!48$$$$T^{14} +$$$$30\!\cdots\!68$$$$p T^{15} +$$$$31\!\cdots\!80$$$$T^{16} +$$$$30\!\cdots\!68$$$$p^{4} T^{17} +$$$$14\!\cdots\!48$$$$p^{6} T^{18} +$$$$11\!\cdots\!24$$$$p^{10} T^{19} +$$$$55\!\cdots\!22$$$$p^{12} T^{20} +$$$$20\!\cdots\!80$$$$p^{15} T^{21} +$$$$18\!\cdots\!52$$$$p^{18} T^{22} +$$$$43\!\cdots\!72$$$$p^{21} T^{23} +$$$$49\!\cdots\!61$$$$p^{24} T^{24} + 61873064288651078904 p^{27} T^{25} + 1073950381186450240 p^{30} T^{26} + 39208048973064 p^{33} T^{27} + 1736544062022 p^{36} T^{28} - 25998216 p^{39} T^{29} + 1866728 p^{42} T^{30} - 48 p^{45} T^{31} + p^{48} T^{32}$$
61 $$1 + 1504 T + 3647736 T^{2} + 4149308096 T^{3} + 5835114087232 T^{4} + 5401405020722688 T^{5} + 5646444277738491752 T^{6} +$$$$44\!\cdots\!44$$$$T^{7} +$$$$37\!\cdots\!68$$$$T^{8} +$$$$25\!\cdots\!32$$$$T^{9} +$$$$18\!\cdots\!04$$$$T^{10} +$$$$11\!\cdots\!12$$$$T^{11} +$$$$73\!\cdots\!48$$$$T^{12} +$$$$39\!\cdots\!16$$$$T^{13} +$$$$22\!\cdots\!48$$$$T^{14} +$$$$11\!\cdots\!76$$$$T^{15} +$$$$57\!\cdots\!94$$$$T^{16} +$$$$11\!\cdots\!76$$$$p^{3} T^{17} +$$$$22\!\cdots\!48$$$$p^{6} T^{18} +$$$$39\!\cdots\!16$$$$p^{9} T^{19} +$$$$73\!\cdots\!48$$$$p^{12} T^{20} +$$$$11\!\cdots\!12$$$$p^{15} T^{21} +$$$$18\!\cdots\!04$$$$p^{18} T^{22} +$$$$25\!\cdots\!32$$$$p^{21} T^{23} +$$$$37\!\cdots\!68$$$$p^{24} T^{24} +$$$$44\!\cdots\!44$$$$p^{27} T^{25} + 5646444277738491752 p^{30} T^{26} + 5401405020722688 p^{33} T^{27} + 5835114087232 p^{36} T^{28} + 4149308096 p^{39} T^{29} + 3647736 p^{42} T^{30} + 1504 p^{45} T^{31} + p^{48} T^{32}$$
67 $$1 - 116 T + 3164788 T^{2} - 225157040 T^{3} + 4954414589886 T^{4} - 176099421633332 T^{5} + 5104447486609003152 T^{6} - 42785692623306968076 T^{7} +$$$$38\!\cdots\!57$$$$T^{8} +$$$$64\!\cdots\!16$$$$p T^{9} +$$$$23\!\cdots\!04$$$$T^{10} +$$$$55\!\cdots\!72$$$$T^{11} +$$$$11\!\cdots\!58$$$$T^{12} +$$$$34\!\cdots\!96$$$$T^{13} +$$$$43\!\cdots\!72$$$$T^{14} +$$$$14\!\cdots\!72$$$$T^{15} +$$$$14\!\cdots\!76$$$$T^{16} +$$$$14\!\cdots\!72$$$$p^{3} T^{17} +$$$$43\!\cdots\!72$$$$p^{6} T^{18} +$$$$34\!\cdots\!96$$$$p^{9} T^{19} +$$$$11\!\cdots\!58$$$$p^{12} T^{20} +$$$$55\!\cdots\!72$$$$p^{15} T^{21} +$$$$23\!\cdots\!04$$$$p^{18} T^{22} +$$$$64\!\cdots\!16$$$$p^{22} T^{23} +$$$$38\!\cdots\!57$$$$p^{24} T^{24} - 42785692623306968076 p^{27} T^{25} + 5104447486609003152 p^{30} T^{26} - 176099421633332 p^{33} T^{27} + 4954414589886 p^{36} T^{28} - 225157040 p^{39} T^{29} + 3164788 p^{42} T^{30} - 116 p^{45} T^{31} + p^{48} T^{32}$$
71 $$1 - 320 T + 3943664 T^{2} - 918758384 T^{3} + 7346862897712 T^{4} - 1090792261595952 T^{5} + 8644471900115051440 T^{6} -$$$$56\!\cdots\!80$$$$T^{7} +$$$$72\!\cdots\!56$$$$T^{8} +$$$$10\!\cdots\!48$$$$T^{9} +$$$$46\!\cdots\!72$$$$T^{10} +$$$$41\!\cdots\!52$$$$T^{11} +$$$$24\!\cdots\!12$$$$T^{12} +$$$$35\!\cdots\!92$$$$T^{13} +$$$$10\!\cdots\!16$$$$T^{14} +$$$$18\!\cdots\!12$$$$T^{15} +$$$$40\!\cdots\!46$$$$T^{16} +$$$$18\!\cdots\!12$$$$p^{3} T^{17} +$$$$10\!\cdots\!16$$$$p^{6} T^{18} +$$$$35\!\cdots\!92$$$$p^{9} T^{19} +$$$$24\!\cdots\!12$$$$p^{12} T^{20} +$$$$41\!\cdots\!52$$$$p^{15} T^{21} +$$$$46\!\cdots\!72$$$$p^{18} T^{22} +$$$$10\!\cdots\!48$$$$p^{21} T^{23} +$$$$72\!\cdots\!56$$$$p^{24} T^{24} -$$$$56\!\cdots\!80$$$$p^{27} T^{25} + 8644471900115051440 p^{30} T^{26} - 1090792261595952 p^{33} T^{27} + 7346862897712 p^{36} T^{28} - 918758384 p^{39} T^{29} + 3943664 p^{42} T^{30} - 320 p^{45} T^{31} + p^{48} T^{32}$$
73 $$1 + 652 T + 3493702 T^{2} + 2440833764 T^{3} + 6129207383826 T^{4} + 4485717947654876 T^{5} + 7259804293940170810 T^{6} +$$$$53\!\cdots\!40$$$$T^{7} +$$$$65\!\cdots\!33$$$$T^{8} +$$$$47\!\cdots\!64$$$$T^{9} +$$$$47\!\cdots\!04$$$$T^{10} +$$$$44\!\cdots\!20$$$$p T^{11} +$$$$28\!\cdots\!54$$$$T^{12} +$$$$18\!\cdots\!72$$$$T^{13} +$$$$14\!\cdots\!52$$$$T^{14} +$$$$84\!\cdots\!00$$$$T^{15} +$$$$60\!\cdots\!36$$$$T^{16} +$$$$84\!\cdots\!00$$$$p^{3} T^{17} +$$$$14\!\cdots\!52$$$$p^{6} T^{18} +$$$$18\!\cdots\!72$$$$p^{9} T^{19} +$$$$28\!\cdots\!54$$$$p^{12} T^{20} +$$$$44\!\cdots\!20$$$$p^{16} T^{21} +$$$$47\!\cdots\!04$$$$p^{18} T^{22} +$$$$47\!\cdots\!64$$$$p^{21} T^{23} +$$$$65\!\cdots\!33$$$$p^{24} T^{24} +$$$$53\!\cdots\!40$$$$p^{27} T^{25} + 7259804293940170810 p^{30} T^{26} + 4485717947654876 p^{33} T^{27} + 6129207383826 p^{36} T^{28} + 2440833764 p^{39} T^{29} + 3493702 p^{42} T^{30} + 652 p^{45} T^{31} + p^{48} T^{32}$$
79 $$1 - 1136 T + 3290504 T^{2} - 3102171312 T^{3} + 5343118359048 T^{4} - 4212460707087024 T^{5} + 5696430130488192600 T^{6} -$$$$38\!\cdots\!00$$$$T^{7} +$$$$46\!\cdots\!56$$$$T^{8} -$$$$28\!\cdots\!44$$$$T^{9} +$$$$31\!\cdots\!12$$$$T^{10} -$$$$17\!\cdots\!32$$$$T^{11} +$$$$19\!\cdots\!28$$$$T^{12} -$$$$10\!\cdots\!48$$$$T^{13} +$$$$11\!\cdots\!16$$$$T^{14} -$$$$58\!\cdots\!24$$$$T^{15} +$$$$59\!\cdots\!98$$$$T^{16} -$$$$58\!\cdots\!24$$$$p^{3} T^{17} +$$$$11\!\cdots\!16$$$$p^{6} T^{18} -$$$$10\!\cdots\!48$$$$p^{9} T^{19} +$$$$19\!\cdots\!28$$$$p^{12} T^{20} -$$$$17\!\cdots\!32$$$$p^{15} T^{21} +$$$$31\!\cdots\!12$$$$p^{18} T^{22} -$$$$28\!\cdots\!44$$$$p^{21} T^{23} +$$$$46\!\cdots\!56$$$$p^{24} T^{24} -$$$$38\!\cdots\!00$$$$p^{27} T^{25} + 5696430130488192600 p^{30} T^{26} - 4212460707087024 p^{33} T^{27} + 5343118359048 p^{36} T^{28} - 3102171312 p^{39} T^{29} + 3290504 p^{42} T^{30} - 1136 p^{45} T^{31} + p^{48} T^{32}$$
83 $$1 + 3300 T + 10551154 T^{2} + 22479136932 T^{3} + 44617340862804 T^{4} + 73062977764417428 T^{5} +$$$$11\!\cdots\!22$$$$T^{6} +$$$$15\!\cdots\!88$$$$T^{7} +$$$$19\!\cdots\!28$$$$T^{8} +$$$$22\!\cdots\!76$$$$T^{9} +$$$$24\!\cdots\!46$$$$T^{10} +$$$$25\!\cdots\!52$$$$T^{11} +$$$$24\!\cdots\!36$$$$T^{12} +$$$$22\!\cdots\!04$$$$T^{13} +$$$$19\!\cdots\!10$$$$T^{14} +$$$$15\!\cdots\!80$$$$T^{15} +$$$$12\!\cdots\!66$$$$T^{16} +$$$$15\!\cdots\!80$$$$p^{3} T^{17} +$$$$19\!\cdots\!10$$$$p^{6} T^{18} +$$$$22\!\cdots\!04$$$$p^{9} T^{19} +$$$$24\!\cdots\!36$$$$p^{12} T^{20} +$$$$25\!\cdots\!52$$$$p^{15} T^{21} +$$$$24\!\cdots\!46$$$$p^{18} T^{22} +$$$$22\!\cdots\!76$$$$p^{21} T^{23} +$$$$19\!\cdots\!28$$$$p^{24} T^{24} +$$$$15\!\cdots\!88$$$$p^{27} T^{25} +$$$$11\!\cdots\!22$$$$p^{30} T^{26} + 73062977764417428 p^{33} T^{27} + 44617340862804 p^{36} T^{28} + 22479136932 p^{39} T^{29} + 10551154 p^{42} T^{30} + 3300 p^{45} T^{31} + p^{48} T^{32}$$
89 $$1 + 2416 T + 8658416 T^{2} + 16549834544 T^{3} + 34057100000480 T^{4} + 52817935586084816 T^{5} + 80644224477541831120 T^{6} +$$$$10\!\cdots\!32$$$$T^{7} +$$$$12\!\cdots\!44$$$$T^{8} +$$$$14\!\cdots\!64$$$$T^{9} +$$$$14\!\cdots\!80$$$$T^{10} +$$$$14\!\cdots\!12$$$$T^{11} +$$$$13\!\cdots\!52$$$$T^{12} +$$$$11\!\cdots\!44$$$$T^{13} +$$$$96\!\cdots\!08$$$$T^{14} +$$$$79\!\cdots\!44$$$$T^{15} +$$$$66\!\cdots\!18$$$$T^{16} +$$$$79\!\cdots\!44$$$$p^{3} T^{17} +$$$$96\!\cdots\!08$$$$p^{6} T^{18} +$$$$11\!\cdots\!44$$$$p^{9} T^{19} +$$$$13\!\cdots\!52$$$$p^{12} T^{20} +$$$$14\!\cdots\!12$$$$p^{15} T^{21} +$$$$14\!\cdots\!80$$$$p^{18} T^{22} +$$$$14\!\cdots\!64$$$$p^{21} T^{23} +$$$$12\!\cdots\!44$$$$p^{24} T^{24} +$$$$10\!\cdots\!32$$$$p^{27} T^{25} + 80644224477541831120 p^{30} T^{26} + 52817935586084816 p^{33} T^{27} + 34057100000480 p^{36} T^{28} + 16549834544 p^{39} T^{29} + 8658416 p^{42} T^{30} + 2416 p^{45} T^{31} + p^{48} T^{32}$$
97 $$1 + 3616 T + 15506868 T^{2} + 38426687008 T^{3} + 98794095726804 T^{4} + 191054549647136160 T^{5} +$$$$37\!\cdots\!96$$$$T^{6} +$$$$59\!\cdots\!92$$$$T^{7} +$$$$95\!\cdots\!08$$$$T^{8} +$$$$13\!\cdots\!12$$$$T^{9} +$$$$18\!\cdots\!60$$$$T^{10} +$$$$22\!\cdots\!52$$$$T^{11} +$$$$27\!\cdots\!80$$$$T^{12} +$$$$30\!\cdots\!84$$$$T^{13} +$$$$33\!\cdots\!28$$$$T^{14} +$$$$33\!\cdots\!48$$$$T^{15} +$$$$33\!\cdots\!74$$$$T^{16} +$$$$33\!\cdots\!48$$$$p^{3} T^{17} +$$$$33\!\cdots\!28$$$$p^{6} T^{18} +$$$$30\!\cdots\!84$$$$p^{9} T^{19} +$$$$27\!\cdots\!80$$$$p^{12} T^{20} +$$$$22\!\cdots\!52$$$$p^{15} T^{21} +$$$$18\!\cdots\!60$$$$p^{18} T^{22} +$$$$13\!\cdots\!12$$$$p^{21} T^{23} +$$$$95\!\cdots\!08$$$$p^{24} T^{24} +$$$$59\!\cdots\!92$$$$p^{27} T^{25} +$$$$37\!\cdots\!96$$$$p^{30} T^{26} + 191054549647136160 p^{33} T^{27} + 98794095726804 p^{36} T^{28} + 38426687008 p^{39} T^{29} + 15506868 p^{42} T^{30} + 3616 p^{45} T^{31} + p^{48} T^{32}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$