# Properties

 Degree $32$ Conductor $6.158\times 10^{28}$ Sign $1$ Motivic weight $3$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 3·2-s + 2·3-s + 15·4-s − 30·5-s − 6·6-s + 56·7-s − 48·8-s − 60·9-s + 90·10-s − 24·11-s + 30·12-s − 68·13-s − 168·14-s − 60·15-s + 202·16-s + 336·17-s + 180·18-s + 352·19-s − 450·20-s + 112·21-s + 72·22-s − 228·23-s − 96·24-s + 828·25-s + 204·26-s − 32·27-s + 840·28-s + ⋯
 L(s)  = 1 − 1.06·2-s + 0.384·3-s + 15/8·4-s − 2.68·5-s − 0.408·6-s + 3.02·7-s − 2.12·8-s − 2.22·9-s + 2.84·10-s − 0.657·11-s + 0.721·12-s − 1.45·13-s − 3.20·14-s − 1.03·15-s + 3.15·16-s + 4.79·17-s + 2.35·18-s + 4.25·19-s − 5.03·20-s + 1.16·21-s + 0.697·22-s − 2.06·23-s − 0.816·24-s + 6.62·25-s + 1.53·26-s − 0.228·27-s + 5.66·28-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$32$$ Conductor: $$3^{32} \cdot 7^{16}$$ Sign: $1$ Motivic weight: $$3$$ Character: induced by $\chi_{63} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(32,\ 3^{32} \cdot 7^{16} ,\ ( \ : [3/2]^{16} ),\ 1 )$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.136302$$ $$L(\frac12)$$ $$\approx$$ $$0.136302$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 - 2 T + 64 T^{2} - 8 p^{3} T^{3} + 703 p T^{4} - 1462 p^{2} T^{5} + 25 p^{7} T^{6} - 5728 p^{4} T^{7} + 5650 p^{5} T^{8} - 5728 p^{7} T^{9} + 25 p^{13} T^{10} - 1462 p^{11} T^{11} + 703 p^{13} T^{12} - 8 p^{18} T^{13} + 64 p^{18} T^{14} - 2 p^{21} T^{15} + p^{24} T^{16}$$
7 $$( 1 - p T + p^{2} T^{2} )^{8}$$
good2 $$1 + 3 T - 3 p T^{2} - 15 T^{3} - 13 T^{4} + 15 T^{5} + 885 T^{6} - 93 p^{2} T^{7} - 1243 p^{3} T^{8} - 1737 p^{2} T^{9} + 5697 T^{10} + 57231 p T^{11} + 70079 p^{2} T^{12} - 12003 p^{7} T^{13} - 161481 p^{4} T^{14} + 114369 p^{6} T^{15} + 354205 p^{6} T^{16} + 114369 p^{9} T^{17} - 161481 p^{10} T^{18} - 12003 p^{16} T^{19} + 70079 p^{14} T^{20} + 57231 p^{16} T^{21} + 5697 p^{18} T^{22} - 1737 p^{23} T^{23} - 1243 p^{27} T^{24} - 93 p^{29} T^{25} + 885 p^{30} T^{26} + 15 p^{33} T^{27} - 13 p^{36} T^{28} - 15 p^{39} T^{29} - 3 p^{43} T^{30} + 3 p^{45} T^{31} + p^{48} T^{32}$$
5 $$1 + 6 p T + 72 T^{2} - 2088 p T^{3} - 166117 T^{4} + 283968 T^{5} + 7725222 p T^{6} + 93427014 p T^{7} - 196342208 T^{8} - 82657218078 T^{9} - 1137104761158 T^{10} - 2696743553724 T^{11} + 27807034365166 p T^{12} + 450188051017926 p T^{13} + 10174499387616861 T^{14} - 175588804032450084 T^{15} - 3536462896978609646 T^{16} - 175588804032450084 p^{3} T^{17} + 10174499387616861 p^{6} T^{18} + 450188051017926 p^{10} T^{19} + 27807034365166 p^{13} T^{20} - 2696743553724 p^{15} T^{21} - 1137104761158 p^{18} T^{22} - 82657218078 p^{21} T^{23} - 196342208 p^{24} T^{24} + 93427014 p^{28} T^{25} + 7725222 p^{31} T^{26} + 283968 p^{33} T^{27} - 166117 p^{36} T^{28} - 2088 p^{40} T^{29} + 72 p^{42} T^{30} + 6 p^{46} T^{31} + p^{48} T^{32}$$
11 $$1 + 24 T - 6312 T^{2} - 112236 T^{3} + 20919641 T^{4} + 234941478 T^{5} - 48781328610 T^{6} - 247581606570 T^{7} + 91374825172267 T^{8} + 19514919797334 T^{9} - 148969481247130356 T^{10} + 376103153664918684 T^{11} + 20252741000401342891 p T^{12} -$$$$61\!\cdots\!80$$$$T^{13} -$$$$31\!\cdots\!40$$$$T^{14} +$$$$35\!\cdots\!36$$$$T^{15} +$$$$42\!\cdots\!85$$$$T^{16} +$$$$35\!\cdots\!36$$$$p^{3} T^{17} -$$$$31\!\cdots\!40$$$$p^{6} T^{18} -$$$$61\!\cdots\!80$$$$p^{9} T^{19} + 20252741000401342891 p^{13} T^{20} + 376103153664918684 p^{15} T^{21} - 148969481247130356 p^{18} T^{22} + 19514919797334 p^{21} T^{23} + 91374825172267 p^{24} T^{24} - 247581606570 p^{27} T^{25} - 48781328610 p^{30} T^{26} + 234941478 p^{33} T^{27} + 20919641 p^{36} T^{28} - 112236 p^{39} T^{29} - 6312 p^{42} T^{30} + 24 p^{45} T^{31} + p^{48} T^{32}$$
13 $$1 + 68 T - 6065 T^{2} - 196604 T^{3} + 34516832 T^{4} - 115057268 T^{5} - 122980759173 T^{6} + 2886205959684 T^{7} + 258268070617101 T^{8} - 13209130095181224 T^{9} - 311290223266507584 T^{10} + 217834710977538120 p^{2} T^{11} -$$$$16\!\cdots\!94$$$$T^{12} -$$$$69\!\cdots\!48$$$$T^{13} +$$$$24\!\cdots\!74$$$$T^{14} +$$$$61\!\cdots\!32$$$$T^{15} -$$$$74\!\cdots\!40$$$$T^{16} +$$$$61\!\cdots\!32$$$$p^{3} T^{17} +$$$$24\!\cdots\!74$$$$p^{6} T^{18} -$$$$69\!\cdots\!48$$$$p^{9} T^{19} -$$$$16\!\cdots\!94$$$$p^{12} T^{20} + 217834710977538120 p^{17} T^{21} - 311290223266507584 p^{18} T^{22} - 13209130095181224 p^{21} T^{23} + 258268070617101 p^{24} T^{24} + 2886205959684 p^{27} T^{25} - 122980759173 p^{30} T^{26} - 115057268 p^{33} T^{27} + 34516832 p^{36} T^{28} - 196604 p^{39} T^{29} - 6065 p^{42} T^{30} + 68 p^{45} T^{31} + p^{48} T^{32}$$
17 $$( 1 - 168 T + 30394 T^{2} - 3297180 T^{3} + 380502793 T^{4} - 32690504880 T^{5} + 3022684670725 T^{6} - 221800606656432 T^{7} + 17377623608249734 T^{8} - 221800606656432 p^{3} T^{9} + 3022684670725 p^{6} T^{10} - 32690504880 p^{9} T^{11} + 380502793 p^{12} T^{12} - 3297180 p^{15} T^{13} + 30394 p^{18} T^{14} - 168 p^{21} T^{15} + p^{24} T^{16} )^{2}$$
19 $$( 1 - 176 T + 41892 T^{2} - 4833442 T^{3} + 719223515 T^{4} - 65014349322 T^{5} + 7631904477739 T^{6} - 579970516177592 T^{7} + 59026078922135409 T^{8} - 579970516177592 p^{3} T^{9} + 7631904477739 p^{6} T^{10} - 65014349322 p^{9} T^{11} + 719223515 p^{12} T^{12} - 4833442 p^{15} T^{13} + 41892 p^{18} T^{14} - 176 p^{21} T^{15} + p^{24} T^{16} )^{2}$$
23 $$1 + 228 T - 25525 T^{2} - 8731368 T^{3} + 389743110 T^{4} + 187567739154 T^{5} - 2655559094585 T^{6} - 2320959795806634 T^{7} + 15687067961480891 T^{8} + 14915354307460983396 T^{9} -$$$$20\!\cdots\!94$$$$T^{10} +$$$$72\!\cdots\!22$$$$T^{11} +$$$$10\!\cdots\!27$$$$T^{12} -$$$$22\!\cdots\!12$$$$T^{13} -$$$$20\!\cdots\!83$$$$T^{14} +$$$$15\!\cdots\!30$$$$T^{15} +$$$$31\!\cdots\!32$$$$T^{16} +$$$$15\!\cdots\!30$$$$p^{3} T^{17} -$$$$20\!\cdots\!83$$$$p^{6} T^{18} -$$$$22\!\cdots\!12$$$$p^{9} T^{19} +$$$$10\!\cdots\!27$$$$p^{12} T^{20} +$$$$72\!\cdots\!22$$$$p^{15} T^{21} -$$$$20\!\cdots\!94$$$$p^{18} T^{22} + 14915354307460983396 p^{21} T^{23} + 15687067961480891 p^{24} T^{24} - 2320959795806634 p^{27} T^{25} - 2655559094585 p^{30} T^{26} + 187567739154 p^{33} T^{27} + 389743110 p^{36} T^{28} - 8731368 p^{39} T^{29} - 25525 p^{42} T^{30} + 228 p^{45} T^{31} + p^{48} T^{32}$$
29 $$1 + 618 T + 78015 T^{2} - 22761822 T^{3} - 3949828339 T^{4} + 1033935290616 T^{5} + 177028776333588 T^{6} - 27273811167129462 T^{7} - 2315043616387851764 T^{8} +$$$$10\!\cdots\!86$$$$T^{9} -$$$$62\!\cdots\!16$$$$T^{10} -$$$$23\!\cdots\!26$$$$T^{11} +$$$$40\!\cdots\!20$$$$T^{12} +$$$$56\!\cdots\!76$$$$T^{13} -$$$$14\!\cdots\!49$$$$T^{14} -$$$$28\!\cdots\!74$$$$T^{15} +$$$$48\!\cdots\!01$$$$T^{16} -$$$$28\!\cdots\!74$$$$p^{3} T^{17} -$$$$14\!\cdots\!49$$$$p^{6} T^{18} +$$$$56\!\cdots\!76$$$$p^{9} T^{19} +$$$$40\!\cdots\!20$$$$p^{12} T^{20} -$$$$23\!\cdots\!26$$$$p^{15} T^{21} -$$$$62\!\cdots\!16$$$$p^{18} T^{22} +$$$$10\!\cdots\!86$$$$p^{21} T^{23} - 2315043616387851764 p^{24} T^{24} - 27273811167129462 p^{27} T^{25} + 177028776333588 p^{30} T^{26} + 1033935290616 p^{33} T^{27} - 3949828339 p^{36} T^{28} - 22761822 p^{39} T^{29} + 78015 p^{42} T^{30} + 618 p^{45} T^{31} + p^{48} T^{32}$$
31 $$1 + 72 T - 122345 T^{2} - 411588 p T^{3} + 7008800553 T^{4} + 937649058354 T^{5} - 246288843940504 T^{6} - 39508523240638446 T^{7} + 6189681260162427926 T^{8} +$$$$11\!\cdots\!06$$$$T^{9} -$$$$13\!\cdots\!30$$$$T^{10} -$$$$31\!\cdots\!70$$$$T^{11} +$$$$21\!\cdots\!42$$$$T^{12} +$$$$78\!\cdots\!78$$$$T^{13} +$$$$23\!\cdots\!09$$$$p T^{14} -$$$$99\!\cdots\!06$$$$T^{15} -$$$$13\!\cdots\!55$$$$T^{16} -$$$$99\!\cdots\!06$$$$p^{3} T^{17} +$$$$23\!\cdots\!09$$$$p^{7} T^{18} +$$$$78\!\cdots\!78$$$$p^{9} T^{19} +$$$$21\!\cdots\!42$$$$p^{12} T^{20} -$$$$31\!\cdots\!70$$$$p^{15} T^{21} -$$$$13\!\cdots\!30$$$$p^{18} T^{22} +$$$$11\!\cdots\!06$$$$p^{21} T^{23} + 6189681260162427926 p^{24} T^{24} - 39508523240638446 p^{27} T^{25} - 246288843940504 p^{30} T^{26} + 937649058354 p^{33} T^{27} + 7008800553 p^{36} T^{28} - 411588 p^{40} T^{29} - 122345 p^{42} T^{30} + 72 p^{45} T^{31} + p^{48} T^{32}$$
37 $$( 1 - 210 T + 273581 T^{2} - 62260500 T^{3} + 37970446276 T^{4} - 8100635347836 T^{5} + 3402478745034716 T^{6} - 624886705874379210 T^{7} +$$$$20\!\cdots\!28$$$$T^{8} - 624886705874379210 p^{3} T^{9} + 3402478745034716 p^{6} T^{10} - 8100635347836 p^{9} T^{11} + 37970446276 p^{12} T^{12} - 62260500 p^{15} T^{13} + 273581 p^{18} T^{14} - 210 p^{21} T^{15} + p^{24} T^{16} )^{2}$$
41 $$1 + 420 T - 287838 T^{2} - 111388608 T^{3} + 54335362343 T^{4} + 15821993821032 T^{5} - 8270886127177332 T^{6} - 1582199089188412056 T^{7} +$$$$10\!\cdots\!33$$$$T^{8} +$$$$12\!\cdots\!60$$$$T^{9} -$$$$10\!\cdots\!02$$$$T^{10} -$$$$76\!\cdots\!28$$$$T^{11} +$$$$94\!\cdots\!59$$$$T^{12} +$$$$33\!\cdots\!24$$$$T^{13} -$$$$74\!\cdots\!76$$$$T^{14} -$$$$72\!\cdots\!96$$$$T^{15} +$$$$54\!\cdots\!89$$$$T^{16} -$$$$72\!\cdots\!96$$$$p^{3} T^{17} -$$$$74\!\cdots\!76$$$$p^{6} T^{18} +$$$$33\!\cdots\!24$$$$p^{9} T^{19} +$$$$94\!\cdots\!59$$$$p^{12} T^{20} -$$$$76\!\cdots\!28$$$$p^{15} T^{21} -$$$$10\!\cdots\!02$$$$p^{18} T^{22} +$$$$12\!\cdots\!60$$$$p^{21} T^{23} +$$$$10\!\cdots\!33$$$$p^{24} T^{24} - 1582199089188412056 p^{27} T^{25} - 8270886127177332 p^{30} T^{26} + 15821993821032 p^{33} T^{27} + 54335362343 p^{36} T^{28} - 111388608 p^{39} T^{29} - 287838 p^{42} T^{30} + 420 p^{45} T^{31} + p^{48} T^{32}$$
43 $$1 - 2 T - 174923 T^{2} - 18126802 T^{3} + 6568829129 T^{4} + 3504431725334 T^{5} + 1175920450315236 T^{6} - 266898513634748244 T^{7} -$$$$13\!\cdots\!38$$$$T^{8} -$$$$25\!\cdots\!48$$$$T^{9} +$$$$33\!\cdots\!62$$$$T^{10} +$$$$20\!\cdots\!40$$$$T^{11} +$$$$30\!\cdots\!22$$$$T^{12} -$$$$22\!\cdots\!90$$$$T^{13} -$$$$31\!\cdots\!27$$$$T^{14} +$$$$89\!\cdots\!28$$$$T^{15} +$$$$20\!\cdots\!33$$$$T^{16} +$$$$89\!\cdots\!28$$$$p^{3} T^{17} -$$$$31\!\cdots\!27$$$$p^{6} T^{18} -$$$$22\!\cdots\!90$$$$p^{9} T^{19} +$$$$30\!\cdots\!22$$$$p^{12} T^{20} +$$$$20\!\cdots\!40$$$$p^{15} T^{21} +$$$$33\!\cdots\!62$$$$p^{18} T^{22} -$$$$25\!\cdots\!48$$$$p^{21} T^{23} -$$$$13\!\cdots\!38$$$$p^{24} T^{24} - 266898513634748244 p^{27} T^{25} + 1175920450315236 p^{30} T^{26} + 3504431725334 p^{33} T^{27} + 6568829129 p^{36} T^{28} - 18126802 p^{39} T^{29} - 174923 p^{42} T^{30} - 2 p^{45} T^{31} + p^{48} T^{32}$$
47 $$1 + 570 T - 504019 T^{2} - 269131554 T^{3} + 178645227255 T^{4} + 75997022312142 T^{5} - 48054772790072906 T^{6} - 15112394704504881348 T^{7} +$$$$10\!\cdots\!88$$$$T^{8} +$$$$22\!\cdots\!32$$$$T^{9} -$$$$18\!\cdots\!92$$$$T^{10} -$$$$26\!\cdots\!90$$$$T^{11} +$$$$28\!\cdots\!88$$$$T^{12} +$$$$22\!\cdots\!72$$$$T^{13} -$$$$37\!\cdots\!29$$$$T^{14} -$$$$88\!\cdots\!70$$$$T^{15} +$$$$41\!\cdots\!29$$$$T^{16} -$$$$88\!\cdots\!70$$$$p^{3} T^{17} -$$$$37\!\cdots\!29$$$$p^{6} T^{18} +$$$$22\!\cdots\!72$$$$p^{9} T^{19} +$$$$28\!\cdots\!88$$$$p^{12} T^{20} -$$$$26\!\cdots\!90$$$$p^{15} T^{21} -$$$$18\!\cdots\!92$$$$p^{18} T^{22} +$$$$22\!\cdots\!32$$$$p^{21} T^{23} +$$$$10\!\cdots\!88$$$$p^{24} T^{24} - 15112394704504881348 p^{27} T^{25} - 48054772790072906 p^{30} T^{26} + 75997022312142 p^{33} T^{27} + 178645227255 p^{36} T^{28} - 269131554 p^{39} T^{29} - 504019 p^{42} T^{30} + 570 p^{45} T^{31} + p^{48} T^{32}$$
53 $$( 1 - 528 T + 603082 T^{2} - 112356498 T^{3} + 113056847713 T^{4} - 1488977365716 T^{5} + 24102465385714279 T^{6} - 3059793563504889246 T^{7} +$$$$52\!\cdots\!50$$$$T^{8} - 3059793563504889246 p^{3} T^{9} + 24102465385714279 p^{6} T^{10} - 1488977365716 p^{9} T^{11} + 113056847713 p^{12} T^{12} - 112356498 p^{15} T^{13} + 603082 p^{18} T^{14} - 528 p^{21} T^{15} + p^{24} T^{16} )^{2}$$
59 $$1 - 150 T - 946117 T^{2} + 200189070 T^{3} + 428040629955 T^{4} - 107278598640996 T^{5} - 129587827131610502 T^{6} + 31309587455023028910 T^{7} +$$$$32\!\cdots\!18$$$$T^{8} -$$$$58\!\cdots\!30$$$$T^{9} -$$$$81\!\cdots\!26$$$$T^{10} +$$$$89\!\cdots\!28$$$$T^{11} +$$$$19\!\cdots\!14$$$$T^{12} -$$$$13\!\cdots\!46$$$$T^{13} -$$$$43\!\cdots\!67$$$$T^{14} +$$$$10\!\cdots\!64$$$$T^{15} +$$$$89\!\cdots\!65$$$$T^{16} +$$$$10\!\cdots\!64$$$$p^{3} T^{17} -$$$$43\!\cdots\!67$$$$p^{6} T^{18} -$$$$13\!\cdots\!46$$$$p^{9} T^{19} +$$$$19\!\cdots\!14$$$$p^{12} T^{20} +$$$$89\!\cdots\!28$$$$p^{15} T^{21} -$$$$81\!\cdots\!26$$$$p^{18} T^{22} -$$$$58\!\cdots\!30$$$$p^{21} T^{23} +$$$$32\!\cdots\!18$$$$p^{24} T^{24} + 31309587455023028910 p^{27} T^{25} - 129587827131610502 p^{30} T^{26} - 107278598640996 p^{33} T^{27} + 428040629955 p^{36} T^{28} + 200189070 p^{39} T^{29} - 946117 p^{42} T^{30} - 150 p^{45} T^{31} + p^{48} T^{32}$$
61 $$1 + 578 T - 1145735 T^{2} - 501446726 T^{3} + 837776641016 T^{4} + 251069047448050 T^{5} - 421088343940353753 T^{6} - 74043629095730306598 T^{7} +$$$$16\!\cdots\!99$$$$T^{8} +$$$$21\!\cdots\!32$$$$p T^{9} -$$$$48\!\cdots\!12$$$$T^{10} -$$$$44\!\cdots\!40$$$$T^{11} +$$$$11\!\cdots\!09$$$$T^{12} -$$$$29\!\cdots\!26$$$$T^{13} -$$$$26\!\cdots\!71$$$$T^{14} +$$$$45\!\cdots\!78$$$$T^{15} +$$$$57\!\cdots\!72$$$$T^{16} +$$$$45\!\cdots\!78$$$$p^{3} T^{17} -$$$$26\!\cdots\!71$$$$p^{6} T^{18} -$$$$29\!\cdots\!26$$$$p^{9} T^{19} +$$$$11\!\cdots\!09$$$$p^{12} T^{20} -$$$$44\!\cdots\!40$$$$p^{15} T^{21} -$$$$48\!\cdots\!12$$$$p^{18} T^{22} +$$$$21\!\cdots\!32$$$$p^{22} T^{23} +$$$$16\!\cdots\!99$$$$p^{24} T^{24} - 74043629095730306598 p^{27} T^{25} - 421088343940353753 p^{30} T^{26} + 251069047448050 p^{33} T^{27} + 837776641016 p^{36} T^{28} - 501446726 p^{39} T^{29} - 1145735 p^{42} T^{30} + 578 p^{45} T^{31} + p^{48} T^{32}$$
67 $$1 - 898 T - 791450 T^{2} + 1047042856 T^{3} + 204780930149 T^{4} - 606381046616276 T^{5} + 43407094416136272 T^{6} +$$$$23\!\cdots\!40$$$$T^{7} -$$$$65\!\cdots\!51$$$$T^{8} -$$$$67\!\cdots\!24$$$$T^{9} +$$$$35\!\cdots\!60$$$$T^{10} +$$$$13\!\cdots\!80$$$$T^{11} -$$$$13\!\cdots\!05$$$$T^{12} -$$$$17\!\cdots\!58$$$$T^{13} +$$$$38\!\cdots\!12$$$$T^{14} +$$$$11\!\cdots\!04$$$$T^{15} -$$$$11\!\cdots\!23$$$$T^{16} +$$$$11\!\cdots\!04$$$$p^{3} T^{17} +$$$$38\!\cdots\!12$$$$p^{6} T^{18} -$$$$17\!\cdots\!58$$$$p^{9} T^{19} -$$$$13\!\cdots\!05$$$$p^{12} T^{20} +$$$$13\!\cdots\!80$$$$p^{15} T^{21} +$$$$35\!\cdots\!60$$$$p^{18} T^{22} -$$$$67\!\cdots\!24$$$$p^{21} T^{23} -$$$$65\!\cdots\!51$$$$p^{24} T^{24} +$$$$23\!\cdots\!40$$$$p^{27} T^{25} + 43407094416136272 p^{30} T^{26} - 606381046616276 p^{33} T^{27} + 204780930149 p^{36} T^{28} + 1047042856 p^{39} T^{29} - 791450 p^{42} T^{30} - 898 p^{45} T^{31} + p^{48} T^{32}$$
71 $$( 1 - 882 T + 1565920 T^{2} - 1172203578 T^{3} + 1342300776931 T^{4} - 886834221035832 T^{5} + 765294077320357237 T^{6} -$$$$44\!\cdots\!26$$$$T^{7} +$$$$31\!\cdots\!95$$$$T^{8} -$$$$44\!\cdots\!26$$$$p^{3} T^{9} + 765294077320357237 p^{6} T^{10} - 886834221035832 p^{9} T^{11} + 1342300776931 p^{12} T^{12} - 1172203578 p^{15} T^{13} + 1565920 p^{18} T^{14} - 882 p^{21} T^{15} + p^{24} T^{16} )^{2}$$
73 $$( 1 - 972 T + 1718351 T^{2} - 1080282636 T^{3} + 1082261782252 T^{4} - 376474331161008 T^{5} + 293371917050729072 T^{6} - 2405960897454683676 T^{7} +$$$$57\!\cdots\!52$$$$T^{8} - 2405960897454683676 p^{3} T^{9} + 293371917050729072 p^{6} T^{10} - 376474331161008 p^{9} T^{11} + 1082261782252 p^{12} T^{12} - 1080282636 p^{15} T^{13} + 1718351 p^{18} T^{14} - 972 p^{21} T^{15} + p^{24} T^{16} )^{2}$$
79 $$1 - 2 p T - 1631978 T^{2} + 326581604 T^{3} + 1003734307967 T^{4} - 276658431385546 T^{5} - 294010705461071412 T^{6} +$$$$16\!\cdots\!94$$$$T^{7} +$$$$10\!\cdots\!84$$$$T^{8} -$$$$77\!\cdots\!36$$$$T^{9} -$$$$89\!\cdots\!86$$$$T^{10} +$$$$15\!\cdots\!56$$$$T^{11} +$$$$53\!\cdots\!98$$$$T^{12} +$$$$92\!\cdots\!86$$$$T^{13} -$$$$20\!\cdots\!95$$$$T^{14} -$$$$39\!\cdots\!96$$$$T^{15} +$$$$81\!\cdots\!30$$$$T^{16} -$$$$39\!\cdots\!96$$$$p^{3} T^{17} -$$$$20\!\cdots\!95$$$$p^{6} T^{18} +$$$$92\!\cdots\!86$$$$p^{9} T^{19} +$$$$53\!\cdots\!98$$$$p^{12} T^{20} +$$$$15\!\cdots\!56$$$$p^{15} T^{21} -$$$$89\!\cdots\!86$$$$p^{18} T^{22} -$$$$77\!\cdots\!36$$$$p^{21} T^{23} +$$$$10\!\cdots\!84$$$$p^{24} T^{24} +$$$$16\!\cdots\!94$$$$p^{27} T^{25} - 294010705461071412 p^{30} T^{26} - 276658431385546 p^{33} T^{27} + 1003734307967 p^{36} T^{28} + 326581604 p^{39} T^{29} - 1631978 p^{42} T^{30} - 2 p^{46} T^{31} + p^{48} T^{32}$$
83 $$1 + 2958 T + 2046194 T^{2} - 1191819780 T^{3} - 156888285003 T^{4} + 2954536317243036 T^{5} + 1354865827302829804 T^{6} -$$$$96\!\cdots\!52$$$$T^{7} +$$$$25\!\cdots\!17$$$$T^{8} +$$$$90\!\cdots\!24$$$$T^{9} -$$$$57\!\cdots\!04$$$$T^{10} -$$$$67\!\cdots\!52$$$$T^{11} +$$$$24\!\cdots\!15$$$$T^{12} -$$$$26\!\cdots\!58$$$$T^{13} -$$$$13\!\cdots\!52$$$$T^{14} +$$$$69\!\cdots\!52$$$$T^{15} +$$$$56\!\cdots\!77$$$$T^{16} +$$$$69\!\cdots\!52$$$$p^{3} T^{17} -$$$$13\!\cdots\!52$$$$p^{6} T^{18} -$$$$26\!\cdots\!58$$$$p^{9} T^{19} +$$$$24\!\cdots\!15$$$$p^{12} T^{20} -$$$$67\!\cdots\!52$$$$p^{15} T^{21} -$$$$57\!\cdots\!04$$$$p^{18} T^{22} +$$$$90\!\cdots\!24$$$$p^{21} T^{23} +$$$$25\!\cdots\!17$$$$p^{24} T^{24} -$$$$96\!\cdots\!52$$$$p^{27} T^{25} + 1354865827302829804 p^{30} T^{26} + 2954536317243036 p^{33} T^{27} - 156888285003 p^{36} T^{28} - 1191819780 p^{39} T^{29} + 2046194 p^{42} T^{30} + 2958 p^{45} T^{31} + p^{48} T^{32}$$
89 $$( 1 - 4380 T + 10823694 T^{2} - 19914754776 T^{3} + 29899723956313 T^{4} - 38260156523638452 T^{5} + 42933634978495882653 T^{6} -$$$$42\!\cdots\!56$$$$T^{7} +$$$$37\!\cdots\!62$$$$T^{8} -$$$$42\!\cdots\!56$$$$p^{3} T^{9} + 42933634978495882653 p^{6} T^{10} - 38260156523638452 p^{9} T^{11} + 29899723956313 p^{12} T^{12} - 19914754776 p^{15} T^{13} + 10823694 p^{18} T^{14} - 4380 p^{21} T^{15} + p^{24} T^{16} )^{2}$$
97 $$1 - 60 T - 3109535 T^{2} + 2828204676 T^{3} + 4865749520445 T^{4} - 7975237084169220 T^{5} - 624844777429079932 T^{6} +$$$$11\!\cdots\!12$$$$T^{7} -$$$$80\!\cdots\!40$$$$T^{8} -$$$$65\!\cdots\!40$$$$T^{9} +$$$$13\!\cdots\!28$$$$T^{10} -$$$$36\!\cdots\!20$$$$T^{11} -$$$$86\!\cdots\!92$$$$T^{12} +$$$$97\!\cdots\!76$$$$T^{13} -$$$$16\!\cdots\!31$$$$T^{14} -$$$$45\!\cdots\!64$$$$T^{15} +$$$$67\!\cdots\!49$$$$T^{16} -$$$$45\!\cdots\!64$$$$p^{3} T^{17} -$$$$16\!\cdots\!31$$$$p^{6} T^{18} +$$$$97\!\cdots\!76$$$$p^{9} T^{19} -$$$$86\!\cdots\!92$$$$p^{12} T^{20} -$$$$36\!\cdots\!20$$$$p^{15} T^{21} +$$$$13\!\cdots\!28$$$$p^{18} T^{22} -$$$$65\!\cdots\!40$$$$p^{21} T^{23} -$$$$80\!\cdots\!40$$$$p^{24} T^{24} +$$$$11\!\cdots\!12$$$$p^{27} T^{25} - 624844777429079932 p^{30} T^{26} - 7975237084169220 p^{33} T^{27} + 4865749520445 p^{36} T^{28} + 2828204676 p^{39} T^{29} - 3109535 p^{42} T^{30} - 60 p^{45} T^{31} + p^{48} T^{32}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−4.15144209598724826996625175124, −3.77458224081294579482075298037, −3.71430661312872125517713898861, −3.60995909498732119252967365325, −3.45121627763063097762332086733, −3.45090934989475781108865918861, −3.43612368671424180778522189786, −3.36674174827280871276542831336, −3.23122089043815030542848849651, −3.18971955606463887201153533825, −3.00260165427286123950953129186, −2.67535155878948797692090943778, −2.51054392916014174353828992722, −2.40098738804147897123503400960, −2.35594366059793525635554377309, −2.20914620181221725350415928445, −2.00542651801110699578233678277, −1.58228085817417117343241623447, −1.57228710194750973228350208005, −1.40194691810868791003134941779, −0.977155801124939165433842024963, −0.855898452752791071067842181422, −0.77697921606178826144602194153, −0.72934243061518344713663861411, −0.03421748582433319819363761749, 0.03421748582433319819363761749, 0.72934243061518344713663861411, 0.77697921606178826144602194153, 0.855898452752791071067842181422, 0.977155801124939165433842024963, 1.40194691810868791003134941779, 1.57228710194750973228350208005, 1.58228085817417117343241623447, 2.00542651801110699578233678277, 2.20914620181221725350415928445, 2.35594366059793525635554377309, 2.40098738804147897123503400960, 2.51054392916014174353828992722, 2.67535155878948797692090943778, 3.00260165427286123950953129186, 3.18971955606463887201153533825, 3.23122089043815030542848849651, 3.36674174827280871276542831336, 3.43612368671424180778522189786, 3.45090934989475781108865918861, 3.45121627763063097762332086733, 3.60995909498732119252967365325, 3.71430661312872125517713898861, 3.77458224081294579482075298037, 4.15144209598724826996625175124

## Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.