Dirichlet series
| L(s) = 1 | + 20·2-s + 3-s + 180·4-s + 20·6-s − 10·7-s + 880·8-s − 18·9-s + 180·12-s − 111·13-s − 200·14-s + 1.76e3·16-s − 90·17-s − 360·18-s − 143·19-s − 10·21-s + 880·24-s − 533·25-s − 2.22e3·26-s − 193·27-s − 1.80e3·28-s − 96·29-s − 6.33e3·32-s − 1.80e3·34-s − 3.24e3·36-s − 2.86e3·38-s − 111·39-s + 537·41-s + ⋯ |
| L(s) = 1 | + 7.07·2-s + 0.192·3-s + 45/2·4-s + 1.36·6-s − 0.539·7-s + 38.8·8-s − 2/3·9-s + 4.33·12-s − 2.36·13-s − 3.81·14-s + 55/2·16-s − 1.28·17-s − 4.71·18-s − 1.72·19-s − 0.103·21-s + 7.48·24-s − 4.26·25-s − 16.7·26-s − 1.37·27-s − 12.1·28-s − 0.614·29-s − 35.0·32-s − 9.07·34-s − 15·36-s − 12.2·38-s − 0.455·39-s + 2.04·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{20} \cdot 19^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{20} \cdot 19^{20}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(2^{20} \cdot 3^{20} \cdot 19^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.59267\times 10^{16}\) |
| Root analytic conductor: | \(2.59349\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 2^{20} \cdot 3^{20} \cdot 19^{20} ,\ ( \ : [3/2]^{20} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(0.5762329028\) |
| \(L(\frac12)\) | \(\approx\) | \(0.5762329028\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - p T + p^{2} T^{2} )^{10} \) |
| 3 | \( 1 - T + 19 T^{2} + 52 p T^{3} - 8 p^{2} T^{4} + 640 p^{2} T^{5} - 335 p^{2} T^{6} + 4721 p^{3} T^{7} + 355 p^{6} T^{8} - 8804 p^{5} T^{9} + 349072 p^{4} T^{10} - 8804 p^{8} T^{11} + 355 p^{12} T^{12} + 4721 p^{12} T^{13} - 335 p^{14} T^{14} + 640 p^{17} T^{15} - 8 p^{20} T^{16} + 52 p^{22} T^{17} + 19 p^{24} T^{18} - p^{27} T^{19} + p^{30} T^{20} \) | |
| 19 | \( 1 + 143 T - 6089 T^{2} - 1933736 T^{3} - 19522123 T^{4} + 465729241 p T^{5} - 249604108 p^{2} T^{6} - 1817483975 p^{3} T^{7} + 42256519778 p^{4} T^{8} + 3170663479 p^{5} T^{9} - 1114119433894 p^{6} T^{10} + 3170663479 p^{8} T^{11} + 42256519778 p^{10} T^{12} - 1817483975 p^{12} T^{13} - 249604108 p^{14} T^{14} + 465729241 p^{16} T^{15} - 19522123 p^{18} T^{16} - 1933736 p^{21} T^{17} - 6089 p^{24} T^{18} + 143 p^{27} T^{19} + p^{30} T^{20} \) | |
| good | 5 | \( 1 + 533 T^{2} + 145714 T^{4} - 304092 T^{5} + 24457387 T^{6} - 133794612 T^{7} + 2480134397 T^{8} - 29233797696 T^{9} + 131810705842 T^{10} - 3627758320704 T^{11} + 162961018762 p T^{12} - 214438706896284 T^{13} + 5558705945008 T^{14} + 1595365421043276 T^{15} + 67954887736748057 T^{16} + 333399268958948316 T^{17} + 1588215410781604481 p T^{18} - \)\(17\!\cdots\!72\)\( T^{19} + \)\(61\!\cdots\!96\)\( T^{20} - \)\(17\!\cdots\!72\)\( p^{3} T^{21} + 1588215410781604481 p^{7} T^{22} + 333399268958948316 p^{9} T^{23} + 67954887736748057 p^{12} T^{24} + 1595365421043276 p^{15} T^{25} + 5558705945008 p^{18} T^{26} - 214438706896284 p^{21} T^{27} + 162961018762 p^{25} T^{28} - 3627758320704 p^{27} T^{29} + 131810705842 p^{30} T^{30} - 29233797696 p^{33} T^{31} + 2480134397 p^{36} T^{32} - 133794612 p^{39} T^{33} + 24457387 p^{42} T^{34} - 304092 p^{45} T^{35} + 145714 p^{48} T^{36} + 533 p^{54} T^{38} + p^{60} T^{40} \) |
| 7 | \( ( 1 + 5 T + 1588 T^{2} + 11131 T^{3} + 1398602 T^{4} + 9977893 T^{5} + 870237542 T^{6} + 833175835 p T^{7} + 414824014505 T^{8} + 2535661256998 T^{9} + 158270639673956 T^{10} + 2535661256998 p^{3} T^{11} + 414824014505 p^{6} T^{12} + 833175835 p^{10} T^{13} + 870237542 p^{12} T^{14} + 9977893 p^{15} T^{15} + 1398602 p^{18} T^{16} + 11131 p^{21} T^{17} + 1588 p^{24} T^{18} + 5 p^{27} T^{19} + p^{30} T^{20} )^{2} \) | |
| 11 | \( 1 - 8467 T^{2} + 41635867 T^{4} - 148822464116 T^{6} + 428194921748795 T^{8} - 1039081614773871143 T^{10} + \)\(21\!\cdots\!24\)\( T^{12} - \)\(41\!\cdots\!81\)\( T^{14} + \)\(69\!\cdots\!48\)\( T^{16} - \)\(10\!\cdots\!21\)\( T^{18} + \)\(14\!\cdots\!10\)\( T^{20} - \)\(10\!\cdots\!21\)\( p^{6} T^{22} + \)\(69\!\cdots\!48\)\( p^{12} T^{24} - \)\(41\!\cdots\!81\)\( p^{18} T^{26} + \)\(21\!\cdots\!24\)\( p^{24} T^{28} - 1039081614773871143 p^{30} T^{30} + 428194921748795 p^{36} T^{32} - 148822464116 p^{42} T^{34} + 41635867 p^{48} T^{36} - 8467 p^{54} T^{38} + p^{60} T^{40} \) | |
| 13 | \( 1 + 111 T + 18202 T^{2} + 1564545 T^{3} + 158391919 T^{4} + 12218838876 T^{5} + 968817531576 T^{6} + 68777794464228 T^{7} + 4640736962044020 T^{8} + 303921106921700244 T^{9} + 18415626937468646952 T^{10} + \)\(11\!\cdots\!92\)\( T^{11} + \)\(62\!\cdots\!69\)\( T^{12} + \)\(35\!\cdots\!39\)\( T^{13} + \)\(18\!\cdots\!02\)\( T^{14} + \)\(99\!\cdots\!89\)\( T^{15} + \)\(50\!\cdots\!87\)\( T^{16} + \)\(25\!\cdots\!68\)\( T^{17} + \)\(12\!\cdots\!60\)\( T^{18} + \)\(59\!\cdots\!48\)\( T^{19} + \)\(28\!\cdots\!68\)\( T^{20} + \)\(59\!\cdots\!48\)\( p^{3} T^{21} + \)\(12\!\cdots\!60\)\( p^{6} T^{22} + \)\(25\!\cdots\!68\)\( p^{9} T^{23} + \)\(50\!\cdots\!87\)\( p^{12} T^{24} + \)\(99\!\cdots\!89\)\( p^{15} T^{25} + \)\(18\!\cdots\!02\)\( p^{18} T^{26} + \)\(35\!\cdots\!39\)\( p^{21} T^{27} + \)\(62\!\cdots\!69\)\( p^{24} T^{28} + \)\(11\!\cdots\!92\)\( p^{27} T^{29} + 18415626937468646952 p^{30} T^{30} + 303921106921700244 p^{33} T^{31} + 4640736962044020 p^{36} T^{32} + 68777794464228 p^{39} T^{33} + 968817531576 p^{42} T^{34} + 12218838876 p^{45} T^{35} + 158391919 p^{48} T^{36} + 1564545 p^{51} T^{37} + 18202 p^{54} T^{38} + 111 p^{57} T^{39} + p^{60} T^{40} \) | |
| 17 | \( 1 + 90 T + 19631 T^{2} + 1523790 T^{3} + 156162688 T^{4} + 11446225110 T^{5} + 551524593415 T^{6} + 58493427784362 T^{7} + 1619905606102079 T^{8} + 357195256634952612 T^{9} + 22641315349966010770 T^{10} + \)\(23\!\cdots\!92\)\( T^{11} + \)\(23\!\cdots\!80\)\( T^{12} + \)\(11\!\cdots\!12\)\( T^{13} + \)\(13\!\cdots\!90\)\( T^{14} + \)\(49\!\cdots\!72\)\( T^{15} + \)\(54\!\cdots\!77\)\( T^{16} + \)\(34\!\cdots\!22\)\( T^{17} + \)\(22\!\cdots\!17\)\( T^{18} + \)\(25\!\cdots\!26\)\( T^{19} + \)\(10\!\cdots\!28\)\( T^{20} + \)\(25\!\cdots\!26\)\( p^{3} T^{21} + \)\(22\!\cdots\!17\)\( p^{6} T^{22} + \)\(34\!\cdots\!22\)\( p^{9} T^{23} + \)\(54\!\cdots\!77\)\( p^{12} T^{24} + \)\(49\!\cdots\!72\)\( p^{15} T^{25} + \)\(13\!\cdots\!90\)\( p^{18} T^{26} + \)\(11\!\cdots\!12\)\( p^{21} T^{27} + \)\(23\!\cdots\!80\)\( p^{24} T^{28} + \)\(23\!\cdots\!92\)\( p^{27} T^{29} + 22641315349966010770 p^{30} T^{30} + 357195256634952612 p^{33} T^{31} + 1619905606102079 p^{36} T^{32} + 58493427784362 p^{39} T^{33} + 551524593415 p^{42} T^{34} + 11446225110 p^{45} T^{35} + 156162688 p^{48} T^{36} + 1523790 p^{51} T^{37} + 19631 p^{54} T^{38} + 90 p^{57} T^{39} + p^{60} T^{40} \) | |
| 23 | \( 1 + 59969 T^{2} + 1673285110 T^{4} - 15823490868 T^{5} + 30548665198387 T^{6} - 877348732410900 T^{7} + 445653602790561185 T^{8} - 22185218846448873432 T^{9} + \)\(59\!\cdots\!14\)\( T^{10} - \)\(38\!\cdots\!72\)\( T^{11} + \)\(74\!\cdots\!46\)\( T^{12} - \)\(65\!\cdots\!20\)\( T^{13} + \)\(85\!\cdots\!28\)\( T^{14} - \)\(10\!\cdots\!48\)\( T^{15} + \)\(91\!\cdots\!49\)\( T^{16} - \)\(16\!\cdots\!00\)\( T^{17} + \)\(10\!\cdots\!85\)\( T^{18} - \)\(22\!\cdots\!84\)\( T^{19} + \)\(12\!\cdots\!68\)\( T^{20} - \)\(22\!\cdots\!84\)\( p^{3} T^{21} + \)\(10\!\cdots\!85\)\( p^{6} T^{22} - \)\(16\!\cdots\!00\)\( p^{9} T^{23} + \)\(91\!\cdots\!49\)\( p^{12} T^{24} - \)\(10\!\cdots\!48\)\( p^{15} T^{25} + \)\(85\!\cdots\!28\)\( p^{18} T^{26} - \)\(65\!\cdots\!20\)\( p^{21} T^{27} + \)\(74\!\cdots\!46\)\( p^{24} T^{28} - \)\(38\!\cdots\!72\)\( p^{27} T^{29} + \)\(59\!\cdots\!14\)\( p^{30} T^{30} - 22185218846448873432 p^{33} T^{31} + 445653602790561185 p^{36} T^{32} - 877348732410900 p^{39} T^{33} + 30548665198387 p^{42} T^{34} - 15823490868 p^{45} T^{35} + 1673285110 p^{48} T^{36} + 59969 p^{54} T^{38} + p^{60} T^{40} \) | |
| 29 | \( 1 + 96 T - 108791 T^{2} + 12310320 T^{3} + 8529945418 T^{4} - 1731814089192 T^{5} - 212259356545713 T^{6} + 132105182698144872 T^{7} - 5743315332289441095 T^{8} - \)\(46\!\cdots\!96\)\( T^{9} + \)\(93\!\cdots\!86\)\( T^{10} + \)\(66\!\cdots\!96\)\( T^{11} - \)\(40\!\cdots\!06\)\( T^{12} + \)\(34\!\cdots\!20\)\( T^{13} + \)\(93\!\cdots\!68\)\( T^{14} - \)\(21\!\cdots\!52\)\( T^{15} + \)\(14\!\cdots\!13\)\( T^{16} + \)\(62\!\cdots\!88\)\( T^{17} - \)\(76\!\cdots\!19\)\( T^{18} - \)\(63\!\cdots\!60\)\( T^{19} + \)\(28\!\cdots\!24\)\( T^{20} - \)\(63\!\cdots\!60\)\( p^{3} T^{21} - \)\(76\!\cdots\!19\)\( p^{6} T^{22} + \)\(62\!\cdots\!88\)\( p^{9} T^{23} + \)\(14\!\cdots\!13\)\( p^{12} T^{24} - \)\(21\!\cdots\!52\)\( p^{15} T^{25} + \)\(93\!\cdots\!68\)\( p^{18} T^{26} + \)\(34\!\cdots\!20\)\( p^{21} T^{27} - \)\(40\!\cdots\!06\)\( p^{24} T^{28} + \)\(66\!\cdots\!96\)\( p^{27} T^{29} + \)\(93\!\cdots\!86\)\( p^{30} T^{30} - \)\(46\!\cdots\!96\)\( p^{33} T^{31} - 5743315332289441095 p^{36} T^{32} + 132105182698144872 p^{39} T^{33} - 212259356545713 p^{42} T^{34} - 1731814089192 p^{45} T^{35} + 8529945418 p^{48} T^{36} + 12310320 p^{51} T^{37} - 108791 p^{54} T^{38} + 96 p^{57} T^{39} + p^{60} T^{40} \) | |
| 31 | \( 1 - 188561 T^{2} + 19312915870 T^{4} - 1478883941079855 T^{6} + 93788311836060050358 T^{8} - \)\(50\!\cdots\!55\)\( T^{10} + \)\(24\!\cdots\!44\)\( T^{12} - \)\(10\!\cdots\!71\)\( T^{14} + \)\(38\!\cdots\!73\)\( T^{16} - \)\(13\!\cdots\!14\)\( T^{18} + \)\(41\!\cdots\!04\)\( T^{20} - \)\(13\!\cdots\!14\)\( p^{6} T^{22} + \)\(38\!\cdots\!73\)\( p^{12} T^{24} - \)\(10\!\cdots\!71\)\( p^{18} T^{26} + \)\(24\!\cdots\!44\)\( p^{24} T^{28} - \)\(50\!\cdots\!55\)\( p^{30} T^{30} + 93788311836060050358 p^{36} T^{32} - 1478883941079855 p^{42} T^{34} + 19312915870 p^{48} T^{36} - 188561 p^{54} T^{38} + p^{60} T^{40} \) | |
| 37 | \( 1 - 532241 T^{2} + 140897894194 T^{4} - 24691330666254111 T^{6} + \)\(32\!\cdots\!46\)\( T^{8} - \)\(33\!\cdots\!75\)\( T^{10} + \)\(29\!\cdots\!20\)\( T^{12} - \)\(21\!\cdots\!27\)\( T^{14} + \)\(14\!\cdots\!05\)\( T^{16} - \)\(84\!\cdots\!22\)\( T^{18} + \)\(44\!\cdots\!12\)\( T^{20} - \)\(84\!\cdots\!22\)\( p^{6} T^{22} + \)\(14\!\cdots\!05\)\( p^{12} T^{24} - \)\(21\!\cdots\!27\)\( p^{18} T^{26} + \)\(29\!\cdots\!20\)\( p^{24} T^{28} - \)\(33\!\cdots\!75\)\( p^{30} T^{30} + \)\(32\!\cdots\!46\)\( p^{36} T^{32} - 24691330666254111 p^{42} T^{34} + 140897894194 p^{48} T^{36} - 532241 p^{54} T^{38} + p^{60} T^{40} \) | |
| 41 | \( 1 - 537 T - 49403 T^{2} + 81280986 T^{3} - 9369555029 T^{4} - 2773171729863 T^{5} + 423961455862542 T^{6} - 170876017416795795 T^{7} + 89750584914366255888 T^{8} + \)\(14\!\cdots\!25\)\( T^{9} - \)\(11\!\cdots\!10\)\( T^{10} - \)\(50\!\cdots\!43\)\( p T^{11} + \)\(47\!\cdots\!77\)\( T^{12} + \)\(13\!\cdots\!96\)\( T^{13} + \)\(11\!\cdots\!63\)\( T^{14} - \)\(73\!\cdots\!69\)\( T^{15} - \)\(27\!\cdots\!89\)\( T^{16} + \)\(10\!\cdots\!58\)\( T^{17} + \)\(99\!\cdots\!36\)\( T^{18} - \)\(42\!\cdots\!18\)\( T^{19} + \)\(10\!\cdots\!24\)\( T^{20} - \)\(42\!\cdots\!18\)\( p^{3} T^{21} + \)\(99\!\cdots\!36\)\( p^{6} T^{22} + \)\(10\!\cdots\!58\)\( p^{9} T^{23} - \)\(27\!\cdots\!89\)\( p^{12} T^{24} - \)\(73\!\cdots\!69\)\( p^{15} T^{25} + \)\(11\!\cdots\!63\)\( p^{18} T^{26} + \)\(13\!\cdots\!96\)\( p^{21} T^{27} + \)\(47\!\cdots\!77\)\( p^{24} T^{28} - \)\(50\!\cdots\!43\)\( p^{28} T^{29} - \)\(11\!\cdots\!10\)\( p^{30} T^{30} + \)\(14\!\cdots\!25\)\( p^{33} T^{31} + 89750584914366255888 p^{36} T^{32} - 170876017416795795 p^{39} T^{33} + 423961455862542 p^{42} T^{34} - 2773171729863 p^{45} T^{35} - 9369555029 p^{48} T^{36} + 81280986 p^{51} T^{37} - 49403 p^{54} T^{38} - 537 p^{57} T^{39} + p^{60} T^{40} \) | |
| 43 | \( 1 - 571 T - 155850 T^{2} + 53603903 T^{3} + 45670544149 T^{4} + 542722047732 T^{5} - 6147099256417230 T^{6} - 973820595445234098 T^{7} + \)\(29\!\cdots\!08\)\( T^{8} + \)\(11\!\cdots\!38\)\( T^{9} + \)\(94\!\cdots\!62\)\( T^{10} - \)\(14\!\cdots\!60\)\( T^{11} - \)\(34\!\cdots\!57\)\( T^{12} - \)\(76\!\cdots\!25\)\( T^{13} - \)\(26\!\cdots\!66\)\( T^{14} + \)\(33\!\cdots\!05\)\( T^{15} + \)\(30\!\cdots\!35\)\( T^{16} + \)\(47\!\cdots\!92\)\( T^{17} - \)\(89\!\cdots\!32\)\( T^{18} - \)\(30\!\cdots\!08\)\( T^{19} - \)\(26\!\cdots\!16\)\( T^{20} - \)\(30\!\cdots\!08\)\( p^{3} T^{21} - \)\(89\!\cdots\!32\)\( p^{6} T^{22} + \)\(47\!\cdots\!92\)\( p^{9} T^{23} + \)\(30\!\cdots\!35\)\( p^{12} T^{24} + \)\(33\!\cdots\!05\)\( p^{15} T^{25} - \)\(26\!\cdots\!66\)\( p^{18} T^{26} - \)\(76\!\cdots\!25\)\( p^{21} T^{27} - \)\(34\!\cdots\!57\)\( p^{24} T^{28} - \)\(14\!\cdots\!60\)\( p^{27} T^{29} + \)\(94\!\cdots\!62\)\( p^{30} T^{30} + \)\(11\!\cdots\!38\)\( p^{33} T^{31} + \)\(29\!\cdots\!08\)\( p^{36} T^{32} - 973820595445234098 p^{39} T^{33} - 6147099256417230 p^{42} T^{34} + 542722047732 p^{45} T^{35} + 45670544149 p^{48} T^{36} + 53603903 p^{51} T^{37} - 155850 p^{54} T^{38} - 571 p^{57} T^{39} + p^{60} T^{40} \) | |
| 47 | \( 1 + 126 T + 745469 T^{2} + 93262302 T^{3} + 292710879274 T^{4} + 38538223353990 T^{5} + 79560507936036931 T^{6} + 11594757598962664518 T^{7} + \)\(16\!\cdots\!21\)\( T^{8} + \)\(27\!\cdots\!60\)\( T^{9} + \)\(28\!\cdots\!18\)\( T^{10} + \)\(56\!\cdots\!32\)\( T^{11} + \)\(41\!\cdots\!62\)\( T^{12} + \)\(98\!\cdots\!60\)\( T^{13} + \)\(52\!\cdots\!92\)\( T^{14} + \)\(14\!\cdots\!52\)\( T^{15} + \)\(59\!\cdots\!49\)\( T^{16} + \)\(19\!\cdots\!58\)\( T^{17} + \)\(63\!\cdots\!29\)\( T^{18} + \)\(23\!\cdots\!14\)\( T^{19} + \)\(66\!\cdots\!44\)\( T^{20} + \)\(23\!\cdots\!14\)\( p^{3} T^{21} + \)\(63\!\cdots\!29\)\( p^{6} T^{22} + \)\(19\!\cdots\!58\)\( p^{9} T^{23} + \)\(59\!\cdots\!49\)\( p^{12} T^{24} + \)\(14\!\cdots\!52\)\( p^{15} T^{25} + \)\(52\!\cdots\!92\)\( p^{18} T^{26} + \)\(98\!\cdots\!60\)\( p^{21} T^{27} + \)\(41\!\cdots\!62\)\( p^{24} T^{28} + \)\(56\!\cdots\!32\)\( p^{27} T^{29} + \)\(28\!\cdots\!18\)\( p^{30} T^{30} + \)\(27\!\cdots\!60\)\( p^{33} T^{31} + \)\(16\!\cdots\!21\)\( p^{36} T^{32} + 11594757598962664518 p^{39} T^{33} + 79560507936036931 p^{42} T^{34} + 38538223353990 p^{45} T^{35} + 292710879274 p^{48} T^{36} + 93262302 p^{51} T^{37} + 745469 p^{54} T^{38} + 126 p^{57} T^{39} + p^{60} T^{40} \) | |
| 53 | \( 1 - 126 T - 726797 T^{2} + 148578714 T^{3} + 226655862976 T^{4} - 59828617069974 T^{5} - 43774400350635117 T^{6} + 9700960111503463782 T^{7} + \)\(83\!\cdots\!59\)\( T^{8} - \)\(30\!\cdots\!68\)\( T^{9} - \)\(20\!\cdots\!74\)\( T^{10} - \)\(19\!\cdots\!64\)\( T^{11} + \)\(41\!\cdots\!28\)\( T^{12} - \)\(17\!\cdots\!32\)\( T^{13} - \)\(56\!\cdots\!02\)\( T^{14} + \)\(12\!\cdots\!24\)\( T^{15} + \)\(68\!\cdots\!61\)\( T^{16} + \)\(77\!\cdots\!38\)\( T^{17} - \)\(12\!\cdots\!63\)\( T^{18} - \)\(91\!\cdots\!62\)\( T^{19} + \)\(23\!\cdots\!60\)\( T^{20} - \)\(91\!\cdots\!62\)\( p^{3} T^{21} - \)\(12\!\cdots\!63\)\( p^{6} T^{22} + \)\(77\!\cdots\!38\)\( p^{9} T^{23} + \)\(68\!\cdots\!61\)\( p^{12} T^{24} + \)\(12\!\cdots\!24\)\( p^{15} T^{25} - \)\(56\!\cdots\!02\)\( p^{18} T^{26} - \)\(17\!\cdots\!32\)\( p^{21} T^{27} + \)\(41\!\cdots\!28\)\( p^{24} T^{28} - \)\(19\!\cdots\!64\)\( p^{27} T^{29} - \)\(20\!\cdots\!74\)\( p^{30} T^{30} - \)\(30\!\cdots\!68\)\( p^{33} T^{31} + \)\(83\!\cdots\!59\)\( p^{36} T^{32} + 9700960111503463782 p^{39} T^{33} - 43774400350635117 p^{42} T^{34} - 59828617069974 p^{45} T^{35} + 226655862976 p^{48} T^{36} + 148578714 p^{51} T^{37} - 726797 p^{54} T^{38} - 126 p^{57} T^{39} + p^{60} T^{40} \) | |
| 59 | \( 1 + 1383 T + 79279 T^{2} - 736199268 T^{3} - 280581855839 T^{4} + 143250371677899 T^{5} + 83795244392071254 T^{6} - 21108152697006890331 T^{7} - \)\(12\!\cdots\!94\)\( T^{8} + \)\(70\!\cdots\!07\)\( T^{9} + \)\(37\!\cdots\!46\)\( T^{10} - \)\(14\!\cdots\!43\)\( T^{11} - \)\(12\!\cdots\!41\)\( T^{12} - \)\(39\!\cdots\!86\)\( T^{13} + \)\(18\!\cdots\!37\)\( T^{14} + \)\(40\!\cdots\!09\)\( T^{15} - \)\(60\!\cdots\!85\)\( T^{16} + \)\(34\!\cdots\!46\)\( T^{17} + \)\(33\!\cdots\!44\)\( T^{18} - \)\(12\!\cdots\!18\)\( T^{19} - \)\(15\!\cdots\!28\)\( T^{20} - \)\(12\!\cdots\!18\)\( p^{3} T^{21} + \)\(33\!\cdots\!44\)\( p^{6} T^{22} + \)\(34\!\cdots\!46\)\( p^{9} T^{23} - \)\(60\!\cdots\!85\)\( p^{12} T^{24} + \)\(40\!\cdots\!09\)\( p^{15} T^{25} + \)\(18\!\cdots\!37\)\( p^{18} T^{26} - \)\(39\!\cdots\!86\)\( p^{21} T^{27} - \)\(12\!\cdots\!41\)\( p^{24} T^{28} - \)\(14\!\cdots\!43\)\( p^{27} T^{29} + \)\(37\!\cdots\!46\)\( p^{30} T^{30} + \)\(70\!\cdots\!07\)\( p^{33} T^{31} - \)\(12\!\cdots\!94\)\( p^{36} T^{32} - 21108152697006890331 p^{39} T^{33} + 83795244392071254 p^{42} T^{34} + 143250371677899 p^{45} T^{35} - 280581855839 p^{48} T^{36} - 736199268 p^{51} T^{37} + 79279 p^{54} T^{38} + 1383 p^{57} T^{39} + p^{60} T^{40} \) | |
| 61 | \( 1 - 149 T - 1224546 T^{2} + 72448113 T^{3} + 791584501965 T^{4} + 14946830357934 T^{5} - 325599411403561580 T^{6} - 33874418344039573274 T^{7} + \)\(89\!\cdots\!28\)\( T^{8} + \)\(20\!\cdots\!70\)\( T^{9} - \)\(15\!\cdots\!32\)\( T^{10} - \)\(78\!\cdots\!86\)\( T^{11} + \)\(70\!\cdots\!23\)\( T^{12} + \)\(22\!\cdots\!07\)\( T^{13} + \)\(37\!\cdots\!66\)\( T^{14} - \)\(47\!\cdots\!47\)\( T^{15} - \)\(10\!\cdots\!57\)\( T^{16} + \)\(76\!\cdots\!08\)\( T^{17} + \)\(11\!\cdots\!80\)\( T^{18} - \)\(61\!\cdots\!16\)\( T^{19} - \)\(66\!\cdots\!20\)\( T^{20} - \)\(61\!\cdots\!16\)\( p^{3} T^{21} + \)\(11\!\cdots\!80\)\( p^{6} T^{22} + \)\(76\!\cdots\!08\)\( p^{9} T^{23} - \)\(10\!\cdots\!57\)\( p^{12} T^{24} - \)\(47\!\cdots\!47\)\( p^{15} T^{25} + \)\(37\!\cdots\!66\)\( p^{18} T^{26} + \)\(22\!\cdots\!07\)\( p^{21} T^{27} + \)\(70\!\cdots\!23\)\( p^{24} T^{28} - \)\(78\!\cdots\!86\)\( p^{27} T^{29} - \)\(15\!\cdots\!32\)\( p^{30} T^{30} + \)\(20\!\cdots\!70\)\( p^{33} T^{31} + \)\(89\!\cdots\!28\)\( p^{36} T^{32} - 33874418344039573274 p^{39} T^{33} - 325599411403561580 p^{42} T^{34} + 14946830357934 p^{45} T^{35} + 791584501965 p^{48} T^{36} + 72448113 p^{51} T^{37} - 1224546 p^{54} T^{38} - 149 p^{57} T^{39} + p^{60} T^{40} \) | |
| 67 | \( 1 + 1626 T + 2715331 T^{2} + 2982147414 T^{3} + 3021667087258 T^{4} + 2526436598191542 T^{5} + 1948612773108323397 T^{6} + \)\(13\!\cdots\!26\)\( T^{7} + \)\(87\!\cdots\!14\)\( T^{8} + \)\(53\!\cdots\!82\)\( T^{9} + \)\(31\!\cdots\!97\)\( T^{10} + \)\(18\!\cdots\!22\)\( T^{11} + \)\(10\!\cdots\!64\)\( T^{12} + \)\(61\!\cdots\!46\)\( T^{13} + \)\(34\!\cdots\!85\)\( T^{14} + \)\(19\!\cdots\!78\)\( T^{15} + \)\(10\!\cdots\!13\)\( T^{16} + \)\(58\!\cdots\!12\)\( T^{17} + \)\(32\!\cdots\!46\)\( T^{18} + \)\(17\!\cdots\!76\)\( T^{19} + \)\(97\!\cdots\!76\)\( T^{20} + \)\(17\!\cdots\!76\)\( p^{3} T^{21} + \)\(32\!\cdots\!46\)\( p^{6} T^{22} + \)\(58\!\cdots\!12\)\( p^{9} T^{23} + \)\(10\!\cdots\!13\)\( p^{12} T^{24} + \)\(19\!\cdots\!78\)\( p^{15} T^{25} + \)\(34\!\cdots\!85\)\( p^{18} T^{26} + \)\(61\!\cdots\!46\)\( p^{21} T^{27} + \)\(10\!\cdots\!64\)\( p^{24} T^{28} + \)\(18\!\cdots\!22\)\( p^{27} T^{29} + \)\(31\!\cdots\!97\)\( p^{30} T^{30} + \)\(53\!\cdots\!82\)\( p^{33} T^{31} + \)\(87\!\cdots\!14\)\( p^{36} T^{32} + \)\(13\!\cdots\!26\)\( p^{39} T^{33} + 1948612773108323397 p^{42} T^{34} + 2526436598191542 p^{45} T^{35} + 3021667087258 p^{48} T^{36} + 2982147414 p^{51} T^{37} + 2715331 p^{54} T^{38} + 1626 p^{57} T^{39} + p^{60} T^{40} \) | |
| 71 | \( 1 + 1368 T - 1046081 T^{2} - 2146822704 T^{3} + 338563044532 T^{4} + 1503641116527984 T^{5} - 151305594796897413 T^{6} - \)\(75\!\cdots\!28\)\( T^{7} + \)\(21\!\cdots\!31\)\( T^{8} + \)\(37\!\cdots\!36\)\( T^{9} - \)\(18\!\cdots\!62\)\( T^{10} - \)\(17\!\cdots\!04\)\( T^{11} + \)\(15\!\cdots\!72\)\( p T^{12} + \)\(67\!\cdots\!56\)\( T^{13} - \)\(59\!\cdots\!02\)\( T^{14} - \)\(22\!\cdots\!08\)\( T^{15} + \)\(28\!\cdots\!09\)\( T^{16} + \)\(62\!\cdots\!32\)\( T^{17} - \)\(11\!\cdots\!91\)\( T^{18} - \)\(12\!\cdots\!44\)\( p T^{19} + \)\(83\!\cdots\!68\)\( p^{2} T^{20} - \)\(12\!\cdots\!44\)\( p^{4} T^{21} - \)\(11\!\cdots\!91\)\( p^{6} T^{22} + \)\(62\!\cdots\!32\)\( p^{9} T^{23} + \)\(28\!\cdots\!09\)\( p^{12} T^{24} - \)\(22\!\cdots\!08\)\( p^{15} T^{25} - \)\(59\!\cdots\!02\)\( p^{18} T^{26} + \)\(67\!\cdots\!56\)\( p^{21} T^{27} + \)\(15\!\cdots\!72\)\( p^{25} T^{28} - \)\(17\!\cdots\!04\)\( p^{27} T^{29} - \)\(18\!\cdots\!62\)\( p^{30} T^{30} + \)\(37\!\cdots\!36\)\( p^{33} T^{31} + \)\(21\!\cdots\!31\)\( p^{36} T^{32} - \)\(75\!\cdots\!28\)\( p^{39} T^{33} - 151305594796897413 p^{42} T^{34} + 1503641116527984 p^{45} T^{35} + 338563044532 p^{48} T^{36} - 2146822704 p^{51} T^{37} - 1046081 p^{54} T^{38} + 1368 p^{57} T^{39} + p^{60} T^{40} \) | |
| 73 | \( 1 - 946 T - 2251263 T^{2} + 2162426558 T^{3} + 2940334917790 T^{4} - 2783227499160318 T^{5} - 2713427049754401573 T^{6} + \)\(25\!\cdots\!70\)\( T^{7} + \)\(19\!\cdots\!90\)\( T^{8} - \)\(18\!\cdots\!14\)\( T^{9} - \)\(10\!\cdots\!25\)\( T^{10} + \)\(10\!\cdots\!62\)\( T^{11} + \)\(44\!\cdots\!24\)\( T^{12} - \)\(49\!\cdots\!86\)\( T^{13} - \)\(14\!\cdots\!61\)\( T^{14} + \)\(19\!\cdots\!94\)\( T^{15} + \)\(27\!\cdots\!09\)\( T^{16} - \)\(55\!\cdots\!32\)\( T^{17} - \)\(29\!\cdots\!22\)\( T^{18} + \)\(79\!\cdots\!44\)\( T^{19} - \)\(15\!\cdots\!16\)\( T^{20} + \)\(79\!\cdots\!44\)\( p^{3} T^{21} - \)\(29\!\cdots\!22\)\( p^{6} T^{22} - \)\(55\!\cdots\!32\)\( p^{9} T^{23} + \)\(27\!\cdots\!09\)\( p^{12} T^{24} + \)\(19\!\cdots\!94\)\( p^{15} T^{25} - \)\(14\!\cdots\!61\)\( p^{18} T^{26} - \)\(49\!\cdots\!86\)\( p^{21} T^{27} + \)\(44\!\cdots\!24\)\( p^{24} T^{28} + \)\(10\!\cdots\!62\)\( p^{27} T^{29} - \)\(10\!\cdots\!25\)\( p^{30} T^{30} - \)\(18\!\cdots\!14\)\( p^{33} T^{31} + \)\(19\!\cdots\!90\)\( p^{36} T^{32} + \)\(25\!\cdots\!70\)\( p^{39} T^{33} - 2713427049754401573 p^{42} T^{34} - 2783227499160318 p^{45} T^{35} + 2940334917790 p^{48} T^{36} + 2162426558 p^{51} T^{37} - 2251263 p^{54} T^{38} - 946 p^{57} T^{39} + p^{60} T^{40} \) | |
| 79 | \( 1 + 2109 T + 3809464 T^{2} + 4907299233 T^{3} + 5242499817463 T^{4} + 4110741386271756 T^{5} + 2258469943640228244 T^{6} + 56251062726674179752 T^{7} - \)\(15\!\cdots\!96\)\( T^{8} - \)\(23\!\cdots\!68\)\( T^{9} - \)\(20\!\cdots\!00\)\( T^{10} - \)\(11\!\cdots\!88\)\( T^{11} - \)\(22\!\cdots\!23\)\( T^{12} + \)\(47\!\cdots\!37\)\( T^{13} + \)\(75\!\cdots\!64\)\( T^{14} + \)\(67\!\cdots\!85\)\( T^{15} + \)\(39\!\cdots\!79\)\( T^{16} + \)\(11\!\cdots\!32\)\( T^{17} - \)\(81\!\cdots\!96\)\( T^{18} - \)\(14\!\cdots\!20\)\( T^{19} - \)\(13\!\cdots\!36\)\( T^{20} - \)\(14\!\cdots\!20\)\( p^{3} T^{21} - \)\(81\!\cdots\!96\)\( p^{6} T^{22} + \)\(11\!\cdots\!32\)\( p^{9} T^{23} + \)\(39\!\cdots\!79\)\( p^{12} T^{24} + \)\(67\!\cdots\!85\)\( p^{15} T^{25} + \)\(75\!\cdots\!64\)\( p^{18} T^{26} + \)\(47\!\cdots\!37\)\( p^{21} T^{27} - \)\(22\!\cdots\!23\)\( p^{24} T^{28} - \)\(11\!\cdots\!88\)\( p^{27} T^{29} - \)\(20\!\cdots\!00\)\( p^{30} T^{30} - \)\(23\!\cdots\!68\)\( p^{33} T^{31} - \)\(15\!\cdots\!96\)\( p^{36} T^{32} + 56251062726674179752 p^{39} T^{33} + 2258469943640228244 p^{42} T^{34} + 4110741386271756 p^{45} T^{35} + 5242499817463 p^{48} T^{36} + 4907299233 p^{51} T^{37} + 3809464 p^{54} T^{38} + 2109 p^{57} T^{39} + p^{60} T^{40} \) | |
| 83 | \( 1 - 5993107 T^{2} + 18525232345675 T^{4} - 38950935000519270236 T^{6} + \)\(62\!\cdots\!59\)\( T^{8} - \)\(80\!\cdots\!43\)\( T^{10} + \)\(86\!\cdots\!28\)\( T^{12} - \)\(78\!\cdots\!21\)\( T^{14} + \)\(62\!\cdots\!12\)\( T^{16} - \)\(43\!\cdots\!93\)\( T^{18} + \)\(26\!\cdots\!58\)\( T^{20} - \)\(43\!\cdots\!93\)\( p^{6} T^{22} + \)\(62\!\cdots\!12\)\( p^{12} T^{24} - \)\(78\!\cdots\!21\)\( p^{18} T^{26} + \)\(86\!\cdots\!28\)\( p^{24} T^{28} - \)\(80\!\cdots\!43\)\( p^{30} T^{30} + \)\(62\!\cdots\!59\)\( p^{36} T^{32} - 38950935000519270236 p^{42} T^{34} + 18525232345675 p^{48} T^{36} - 5993107 p^{54} T^{38} + p^{60} T^{40} \) | |
| 89 | \( 1 + 1938 T - 2929745 T^{2} - 5238651054 T^{3} + 9177887165428 T^{4} + 10037249134671186 T^{5} - 18748691683479232197 T^{6} - \)\(10\!\cdots\!98\)\( T^{7} + \)\(30\!\cdots\!63\)\( T^{8} + \)\(50\!\cdots\!72\)\( T^{9} - \)\(36\!\cdots\!82\)\( T^{10} + \)\(58\!\cdots\!84\)\( T^{11} + \)\(33\!\cdots\!44\)\( T^{12} - \)\(14\!\cdots\!08\)\( T^{13} - \)\(22\!\cdots\!26\)\( T^{14} + \)\(16\!\cdots\!24\)\( T^{15} + \)\(10\!\cdots\!45\)\( T^{16} - \)\(10\!\cdots\!98\)\( T^{17} - \)\(16\!\cdots\!63\)\( T^{18} + \)\(32\!\cdots\!78\)\( T^{19} - \)\(40\!\cdots\!88\)\( T^{20} + \)\(32\!\cdots\!78\)\( p^{3} T^{21} - \)\(16\!\cdots\!63\)\( p^{6} T^{22} - \)\(10\!\cdots\!98\)\( p^{9} T^{23} + \)\(10\!\cdots\!45\)\( p^{12} T^{24} + \)\(16\!\cdots\!24\)\( p^{15} T^{25} - \)\(22\!\cdots\!26\)\( p^{18} T^{26} - \)\(14\!\cdots\!08\)\( p^{21} T^{27} + \)\(33\!\cdots\!44\)\( p^{24} T^{28} + \)\(58\!\cdots\!84\)\( p^{27} T^{29} - \)\(36\!\cdots\!82\)\( p^{30} T^{30} + \)\(50\!\cdots\!72\)\( p^{33} T^{31} + \)\(30\!\cdots\!63\)\( p^{36} T^{32} - \)\(10\!\cdots\!98\)\( p^{39} T^{33} - 18748691683479232197 p^{42} T^{34} + 10037249134671186 p^{45} T^{35} + 9177887165428 p^{48} T^{36} - 5238651054 p^{51} T^{37} - 2929745 p^{54} T^{38} + 1938 p^{57} T^{39} + p^{60} T^{40} \) | |
| 97 | \( 1 - 1791 T + 8037463 T^{2} - 12480110676 T^{3} + 32902545078811 T^{4} - 46377007747507719 T^{5} + 91717242179283854886 T^{6} - \)\(12\!\cdots\!39\)\( T^{7} + \)\(19\!\cdots\!08\)\( T^{8} - \)\(24\!\cdots\!59\)\( T^{9} + \)\(34\!\cdots\!62\)\( T^{10} - \)\(40\!\cdots\!27\)\( T^{11} + \)\(50\!\cdots\!37\)\( T^{12} - \)\(56\!\cdots\!98\)\( T^{13} + \)\(64\!\cdots\!81\)\( T^{14} - \)\(70\!\cdots\!15\)\( T^{15} + \)\(73\!\cdots\!31\)\( T^{16} - \)\(78\!\cdots\!22\)\( T^{17} + \)\(76\!\cdots\!12\)\( T^{18} - \)\(78\!\cdots\!50\)\( T^{19} + \)\(72\!\cdots\!68\)\( T^{20} - \)\(78\!\cdots\!50\)\( p^{3} T^{21} + \)\(76\!\cdots\!12\)\( p^{6} T^{22} - \)\(78\!\cdots\!22\)\( p^{9} T^{23} + \)\(73\!\cdots\!31\)\( p^{12} T^{24} - \)\(70\!\cdots\!15\)\( p^{15} T^{25} + \)\(64\!\cdots\!81\)\( p^{18} T^{26} - \)\(56\!\cdots\!98\)\( p^{21} T^{27} + \)\(50\!\cdots\!37\)\( p^{24} T^{28} - \)\(40\!\cdots\!27\)\( p^{27} T^{29} + \)\(34\!\cdots\!62\)\( p^{30} T^{30} - \)\(24\!\cdots\!59\)\( p^{33} T^{31} + \)\(19\!\cdots\!08\)\( p^{36} T^{32} - \)\(12\!\cdots\!39\)\( p^{39} T^{33} + 91717242179283854886 p^{42} T^{34} - 46377007747507719 p^{45} T^{35} + 32902545078811 p^{48} T^{36} - 12480110676 p^{51} T^{37} + 8037463 p^{54} T^{38} - 1791 p^{57} T^{39} + p^{60} T^{40} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.08068200204817454513666862794, −2.98130064172688215775378439232, −2.95888316889863693936706593051, −2.91994661848293985417487307837, −2.89268745438223768927406289656, −2.87396447707439086462755390951, −2.59221684508049874514899356941, −2.45322334049697507182398338755, −2.28496109279502990157423358968, −2.17563190526877716311198427568, −2.10491401272578080581488004370, −2.08121663738766325815107849139, −1.96651904334880604364226230327, −1.87533214535118919141457608874, −1.77468667437681441833106015352, −1.60000451677495312854352856504, −1.59066506390600102054835342782, −1.25257163577508775033350309053, −1.24291530244636335695893365812, −0.75461011928062507767741335273, −0.67456222556268492842457345050, −0.23899567562741293374396200072, −0.22937811862313039883441595706, −0.11306626104939378668865311678, −0.092803406045943900070052304923, 0.092803406045943900070052304923, 0.11306626104939378668865311678, 0.22937811862313039883441595706, 0.23899567562741293374396200072, 0.67456222556268492842457345050, 0.75461011928062507767741335273, 1.24291530244636335695893365812, 1.25257163577508775033350309053, 1.59066506390600102054835342782, 1.60000451677495312854352856504, 1.77468667437681441833106015352, 1.87533214535118919141457608874, 1.96651904334880604364226230327, 2.08121663738766325815107849139, 2.10491401272578080581488004370, 2.17563190526877716311198427568, 2.28496109279502990157423358968, 2.45322334049697507182398338755, 2.59221684508049874514899356941, 2.87396447707439086462755390951, 2.89268745438223768927406289656, 2.91994661848293985417487307837, 2.95888316889863693936706593051, 2.98130064172688215775378439232, 3.08068200204817454513666862794
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.