Dirichlet series
| L(s) = 1 | − 72·3-s + 12·5-s + 40·7-s + 2.70e3·9-s − 108·11-s − 864·15-s − 168·17-s + 948·19-s − 2.88e3·21-s − 768·23-s − 1.47e3·25-s − 6.99e4·27-s − 1.60e3·29-s − 3.21e3·31-s + 7.77e3·33-s + 480·35-s − 1.82e3·37-s + 2.88e3·43-s + 3.24e4·45-s + 744·47-s − 1.09e3·49-s + 1.20e4·51-s + 4.47e3·53-s − 1.29e3·55-s − 6.82e4·57-s − 4.66e3·59-s + 1.77e4·61-s + ⋯ |
| L(s) = 1 | − 8·3-s + 0.479·5-s + 0.816·7-s + 33.3·9-s − 0.892·11-s − 3.83·15-s − 0.581·17-s + 2.62·19-s − 6.53·21-s − 1.45·23-s − 2.35·25-s − 96·27-s − 1.91·29-s − 3.34·31-s + 7.14·33-s + 0.391·35-s − 1.32·37-s + 1.56·43-s + 16·45-s + 0.336·47-s − 0.454·49-s + 4.65·51-s + 1.59·53-s − 0.428·55-s − 21.0·57-s − 1.34·59-s + 4.77·61-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{48} \cdot 3^{16} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.84318\times 10^{19}\) |
| Root analytic conductor: | \(4.16727\) |
| Motivic weight: | \(4\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{48} \cdot 3^{16} \cdot 7^{16} ,\ ( \ : [2]^{16} ),\ 1 )\) |
Particular Values
| \(L(\frac{5}{2})\) | \(\approx\) | \(0.004718833680\) |
| \(L(\frac12)\) | \(\approx\) | \(0.004718833680\) |
| \(L(3)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + p^{2} T + p^{3} T^{2} )^{8} \) | |
| 7 | \( 1 - 40 T + 2692 T^{2} - 32048 p T^{3} + 160442 p^{2} T^{4} - 2513992 p^{3} T^{5} + 16185392 p^{4} T^{6} - 13107704 p^{6} T^{7} + 1006401811 p^{6} T^{8} - 13107704 p^{10} T^{9} + 16185392 p^{12} T^{10} - 2513992 p^{15} T^{11} + 160442 p^{18} T^{12} - 32048 p^{21} T^{13} + 2692 p^{24} T^{14} - 40 p^{28} T^{15} + p^{32} T^{16} \) | |
| good | 5 | \( 1 - 12 T + 1618 T^{2} - 3768 p T^{3} + 1265347 T^{4} - 18127344 T^{5} + 311370926 T^{6} - 1842768804 T^{7} - 288375378007 T^{8} + 9722110835688 T^{9} - 255955181180204 T^{10} + 1509162317887416 p T^{11} - 40209302055607186 T^{12} - 71507446910726736 p^{2} T^{13} + 92017840051341584168 T^{14} - \)\(24\!\cdots\!88\)\( p^{2} T^{15} + \)\(84\!\cdots\!94\)\( T^{16} - \)\(24\!\cdots\!88\)\( p^{6} T^{17} + 92017840051341584168 p^{8} T^{18} - 71507446910726736 p^{14} T^{19} - 40209302055607186 p^{16} T^{20} + 1509162317887416 p^{21} T^{21} - 255955181180204 p^{24} T^{22} + 9722110835688 p^{28} T^{23} - 288375378007 p^{32} T^{24} - 1842768804 p^{36} T^{25} + 311370926 p^{40} T^{26} - 18127344 p^{44} T^{27} + 1265347 p^{48} T^{28} - 3768 p^{53} T^{29} + 1618 p^{56} T^{30} - 12 p^{60} T^{31} + p^{64} T^{32} \) |
| 11 | \( 1 + 108 T - 64358 T^{2} - 6268776 T^{3} + 2086527043 T^{4} + 161967563232 T^{5} - 50613206768074 T^{6} - 2525722228156524 T^{7} + 1115785606242686105 T^{8} + 31530812694702752712 T^{9} - \)\(23\!\cdots\!12\)\( T^{10} - \)\(49\!\cdots\!16\)\( T^{11} + \)\(42\!\cdots\!54\)\( T^{12} + \)\(76\!\cdots\!32\)\( T^{13} - \)\(69\!\cdots\!80\)\( T^{14} - \)\(52\!\cdots\!60\)\( T^{15} + \)\(10\!\cdots\!22\)\( T^{16} - \)\(52\!\cdots\!60\)\( p^{4} T^{17} - \)\(69\!\cdots\!80\)\( p^{8} T^{18} + \)\(76\!\cdots\!32\)\( p^{12} T^{19} + \)\(42\!\cdots\!54\)\( p^{16} T^{20} - \)\(49\!\cdots\!16\)\( p^{20} T^{21} - \)\(23\!\cdots\!12\)\( p^{24} T^{22} + 31530812694702752712 p^{28} T^{23} + 1115785606242686105 p^{32} T^{24} - 2525722228156524 p^{36} T^{25} - 50613206768074 p^{40} T^{26} + 161967563232 p^{44} T^{27} + 2086527043 p^{48} T^{28} - 6268776 p^{52} T^{29} - 64358 p^{56} T^{30} + 108 p^{60} T^{31} + p^{64} T^{32} \) | |
| 13 | \( 1 - 295172 T^{2} + 42183541126 T^{4} - 301256003231272 p T^{6} + \)\(26\!\cdots\!93\)\( T^{8} - \)\(14\!\cdots\!56\)\( T^{10} + \)\(62\!\cdots\!78\)\( T^{12} - \)\(23\!\cdots\!88\)\( T^{14} + \)\(71\!\cdots\!04\)\( T^{16} - \)\(23\!\cdots\!88\)\( p^{8} T^{18} + \)\(62\!\cdots\!78\)\( p^{16} T^{20} - \)\(14\!\cdots\!56\)\( p^{24} T^{22} + \)\(26\!\cdots\!93\)\( p^{32} T^{24} - 301256003231272 p^{41} T^{26} + 42183541126 p^{48} T^{28} - 295172 p^{56} T^{30} + p^{64} T^{32} \) | |
| 17 | \( 1 + 168 T + 22176 p T^{2} + 61754112 T^{3} + 66603583044 T^{4} + 12006282038424 T^{5} + 8255522457992256 T^{6} + 1742540548150538472 T^{7} + \)\(91\!\cdots\!42\)\( T^{8} + \)\(20\!\cdots\!00\)\( T^{9} + \)\(94\!\cdots\!04\)\( T^{10} + \)\(22\!\cdots\!04\)\( T^{11} + \)\(92\!\cdots\!72\)\( T^{12} + \)\(24\!\cdots\!56\)\( T^{13} + \)\(85\!\cdots\!16\)\( T^{14} + \)\(23\!\cdots\!96\)\( T^{15} + \)\(73\!\cdots\!71\)\( T^{16} + \)\(23\!\cdots\!96\)\( p^{4} T^{17} + \)\(85\!\cdots\!16\)\( p^{8} T^{18} + \)\(24\!\cdots\!56\)\( p^{12} T^{19} + \)\(92\!\cdots\!72\)\( p^{16} T^{20} + \)\(22\!\cdots\!04\)\( p^{20} T^{21} + \)\(94\!\cdots\!04\)\( p^{24} T^{22} + \)\(20\!\cdots\!00\)\( p^{28} T^{23} + \)\(91\!\cdots\!42\)\( p^{32} T^{24} + 1742540548150538472 p^{36} T^{25} + 8255522457992256 p^{40} T^{26} + 12006282038424 p^{44} T^{27} + 66603583044 p^{48} T^{28} + 61754112 p^{52} T^{29} + 22176 p^{57} T^{30} + 168 p^{60} T^{31} + p^{64} T^{32} \) | |
| 19 | \( 1 - 948 T + 959554 T^{2} - 625666728 T^{3} + 367505077267 T^{4} - 175929217837584 T^{5} + 78332810735487806 T^{6} - 29988896729173969692 T^{7} + \)\(11\!\cdots\!49\)\( T^{8} - \)\(43\!\cdots\!96\)\( T^{9} + \)\(18\!\cdots\!52\)\( T^{10} - \)\(71\!\cdots\!96\)\( T^{11} + \)\(30\!\cdots\!34\)\( T^{12} - \)\(11\!\cdots\!76\)\( T^{13} + \)\(48\!\cdots\!72\)\( T^{14} - \)\(17\!\cdots\!52\)\( T^{15} + \)\(67\!\cdots\!30\)\( T^{16} - \)\(17\!\cdots\!52\)\( p^{4} T^{17} + \)\(48\!\cdots\!72\)\( p^{8} T^{18} - \)\(11\!\cdots\!76\)\( p^{12} T^{19} + \)\(30\!\cdots\!34\)\( p^{16} T^{20} - \)\(71\!\cdots\!96\)\( p^{20} T^{21} + \)\(18\!\cdots\!52\)\( p^{24} T^{22} - \)\(43\!\cdots\!96\)\( p^{28} T^{23} + \)\(11\!\cdots\!49\)\( p^{32} T^{24} - 29988896729173969692 p^{36} T^{25} + 78332810735487806 p^{40} T^{26} - 175929217837584 p^{44} T^{27} + 367505077267 p^{48} T^{28} - 625666728 p^{52} T^{29} + 959554 p^{56} T^{30} - 948 p^{60} T^{31} + p^{64} T^{32} \) | |
| 23 | \( 1 + 768 T - 694200 T^{2} - 341097984 T^{3} + 362545215428 T^{4} - 14703263638272 T^{5} - 167490297347201040 T^{6} + 36196708055827466496 T^{7} + \)\(49\!\cdots\!86\)\( T^{8} - \)\(13\!\cdots\!88\)\( T^{9} - \)\(10\!\cdots\!60\)\( T^{10} + \)\(47\!\cdots\!76\)\( T^{11} + \)\(20\!\cdots\!88\)\( T^{12} - \)\(13\!\cdots\!12\)\( T^{13} - \)\(24\!\cdots\!60\)\( T^{14} + \)\(17\!\cdots\!04\)\( T^{15} + \)\(91\!\cdots\!19\)\( T^{16} + \)\(17\!\cdots\!04\)\( p^{4} T^{17} - \)\(24\!\cdots\!60\)\( p^{8} T^{18} - \)\(13\!\cdots\!12\)\( p^{12} T^{19} + \)\(20\!\cdots\!88\)\( p^{16} T^{20} + \)\(47\!\cdots\!76\)\( p^{20} T^{21} - \)\(10\!\cdots\!60\)\( p^{24} T^{22} - \)\(13\!\cdots\!88\)\( p^{28} T^{23} + \)\(49\!\cdots\!86\)\( p^{32} T^{24} + 36196708055827466496 p^{36} T^{25} - 167490297347201040 p^{40} T^{26} - 14703263638272 p^{44} T^{27} + 362545215428 p^{48} T^{28} - 341097984 p^{52} T^{29} - 694200 p^{56} T^{30} + 768 p^{60} T^{31} + p^{64} T^{32} \) | |
| 29 | \( ( 1 + 804 T + 2114126 T^{2} + 2033020608 T^{3} + 2303573631249 T^{4} + 1919378813807496 T^{5} + 1790869579766492182 T^{6} + \)\(10\!\cdots\!08\)\( T^{7} + \)\(12\!\cdots\!40\)\( T^{8} + \)\(10\!\cdots\!08\)\( p^{4} T^{9} + 1790869579766492182 p^{8} T^{10} + 1919378813807496 p^{12} T^{11} + 2303573631249 p^{16} T^{12} + 2033020608 p^{20} T^{13} + 2114126 p^{24} T^{14} + 804 p^{28} T^{15} + p^{32} T^{16} )^{2} \) | |
| 31 | \( 1 + 3216 T + 6668668 T^{2} + 10359109056 T^{3} + 12035821413526 T^{4} + 10684983972441168 T^{5} + 6100253189807876600 T^{6} + \)\(12\!\cdots\!44\)\( T^{7} - \)\(10\!\cdots\!87\)\( T^{8} + \)\(94\!\cdots\!20\)\( T^{9} + \)\(68\!\cdots\!72\)\( T^{10} + \)\(13\!\cdots\!04\)\( T^{11} + \)\(16\!\cdots\!82\)\( T^{12} + \)\(45\!\cdots\!00\)\( p T^{13} + \)\(91\!\cdots\!04\)\( T^{14} + \)\(12\!\cdots\!20\)\( p T^{15} + \)\(17\!\cdots\!48\)\( T^{16} + \)\(12\!\cdots\!20\)\( p^{5} T^{17} + \)\(91\!\cdots\!04\)\( p^{8} T^{18} + \)\(45\!\cdots\!00\)\( p^{13} T^{19} + \)\(16\!\cdots\!82\)\( p^{16} T^{20} + \)\(13\!\cdots\!04\)\( p^{20} T^{21} + \)\(68\!\cdots\!72\)\( p^{24} T^{22} + \)\(94\!\cdots\!20\)\( p^{28} T^{23} - \)\(10\!\cdots\!87\)\( p^{32} T^{24} + \)\(12\!\cdots\!44\)\( p^{36} T^{25} + 6100253189807876600 p^{40} T^{26} + 10684983972441168 p^{44} T^{27} + 12035821413526 p^{48} T^{28} + 10359109056 p^{52} T^{29} + 6668668 p^{56} T^{30} + 3216 p^{60} T^{31} + p^{64} T^{32} \) | |
| 37 | \( 1 + 1820 T + 2243386 T^{2} - 4181941160 T^{3} - 19502956004029 T^{4} - 35349817948568288 T^{5} - 23617541014289135658 T^{6} + \)\(50\!\cdots\!84\)\( T^{7} + \)\(19\!\cdots\!85\)\( T^{8} + \)\(27\!\cdots\!00\)\( T^{9} + \)\(13\!\cdots\!16\)\( T^{10} - \)\(40\!\cdots\!76\)\( T^{11} - \)\(11\!\cdots\!58\)\( T^{12} - \)\(13\!\cdots\!40\)\( T^{13} - \)\(21\!\cdots\!80\)\( T^{14} + \)\(21\!\cdots\!12\)\( T^{15} + \)\(45\!\cdots\!94\)\( T^{16} + \)\(21\!\cdots\!12\)\( p^{4} T^{17} - \)\(21\!\cdots\!80\)\( p^{8} T^{18} - \)\(13\!\cdots\!40\)\( p^{12} T^{19} - \)\(11\!\cdots\!58\)\( p^{16} T^{20} - \)\(40\!\cdots\!76\)\( p^{20} T^{21} + \)\(13\!\cdots\!16\)\( p^{24} T^{22} + \)\(27\!\cdots\!00\)\( p^{28} T^{23} + \)\(19\!\cdots\!85\)\( p^{32} T^{24} + \)\(50\!\cdots\!84\)\( p^{36} T^{25} - 23617541014289135658 p^{40} T^{26} - 35349817948568288 p^{44} T^{27} - 19502956004029 p^{48} T^{28} - 4181941160 p^{52} T^{29} + 2243386 p^{56} T^{30} + 1820 p^{60} T^{31} + p^{64} T^{32} \) | |
| 41 | \( 1 - 24073200 T^{2} + 284455058989944 T^{4} - \)\(22\!\cdots\!20\)\( T^{6} + \)\(13\!\cdots\!88\)\( T^{8} - \)\(65\!\cdots\!60\)\( T^{10} + \)\(26\!\cdots\!92\)\( T^{12} - \)\(94\!\cdots\!80\)\( T^{14} + \)\(28\!\cdots\!54\)\( T^{16} - \)\(94\!\cdots\!80\)\( p^{8} T^{18} + \)\(26\!\cdots\!92\)\( p^{16} T^{20} - \)\(65\!\cdots\!60\)\( p^{24} T^{22} + \)\(13\!\cdots\!88\)\( p^{32} T^{24} - \)\(22\!\cdots\!20\)\( p^{40} T^{26} + 284455058989944 p^{48} T^{28} - 24073200 p^{56} T^{30} + p^{64} T^{32} \) | |
| 43 | \( ( 1 - 1444 T + 9006486 T^{2} - 13541633384 T^{3} + 52462733762369 T^{4} - 91650475778297208 T^{5} + \)\(24\!\cdots\!62\)\( T^{6} - \)\(43\!\cdots\!40\)\( T^{7} + \)\(90\!\cdots\!24\)\( T^{8} - \)\(43\!\cdots\!40\)\( p^{4} T^{9} + \)\(24\!\cdots\!62\)\( p^{8} T^{10} - 91650475778297208 p^{12} T^{11} + 52462733762369 p^{16} T^{12} - 13541633384 p^{20} T^{13} + 9006486 p^{24} T^{14} - 1444 p^{28} T^{15} + p^{32} T^{16} )^{2} \) | |
| 47 | \( 1 - 744 T + 22591360 T^{2} - 16670694912 T^{3} + 254976192342340 T^{4} - 91958236927423704 T^{5} + \)\(18\!\cdots\!08\)\( T^{6} + \)\(62\!\cdots\!64\)\( T^{7} + \)\(93\!\cdots\!78\)\( T^{8} + \)\(13\!\cdots\!80\)\( T^{9} + \)\(39\!\cdots\!32\)\( T^{10} + \)\(10\!\cdots\!92\)\( T^{11} + \)\(19\!\cdots\!76\)\( T^{12} + \)\(55\!\cdots\!60\)\( T^{13} + \)\(12\!\cdots\!24\)\( T^{14} + \)\(23\!\cdots\!04\)\( T^{15} + \)\(67\!\cdots\!51\)\( T^{16} + \)\(23\!\cdots\!04\)\( p^{4} T^{17} + \)\(12\!\cdots\!24\)\( p^{8} T^{18} + \)\(55\!\cdots\!60\)\( p^{12} T^{19} + \)\(19\!\cdots\!76\)\( p^{16} T^{20} + \)\(10\!\cdots\!92\)\( p^{20} T^{21} + \)\(39\!\cdots\!32\)\( p^{24} T^{22} + \)\(13\!\cdots\!80\)\( p^{28} T^{23} + \)\(93\!\cdots\!78\)\( p^{32} T^{24} + \)\(62\!\cdots\!64\)\( p^{36} T^{25} + \)\(18\!\cdots\!08\)\( p^{40} T^{26} - 91958236927423704 p^{44} T^{27} + 254976192342340 p^{48} T^{28} - 16670694912 p^{52} T^{29} + 22591360 p^{56} T^{30} - 744 p^{60} T^{31} + p^{64} T^{32} \) | |
| 53 | \( 1 - 4476 T - 17031278 T^{2} + 58882798008 T^{3} + 262036549366003 T^{4} - 284049247837485888 T^{5} - \)\(23\!\cdots\!46\)\( T^{6} - \)\(19\!\cdots\!48\)\( T^{7} + \)\(84\!\cdots\!37\)\( T^{8} + \)\(23\!\cdots\!60\)\( T^{9} + \)\(44\!\cdots\!08\)\( T^{10} + \)\(19\!\cdots\!08\)\( T^{11} + \)\(20\!\cdots\!30\)\( T^{12} - \)\(26\!\cdots\!16\)\( T^{13} - \)\(12\!\cdots\!24\)\( T^{14} + \)\(12\!\cdots\!72\)\( T^{15} + \)\(15\!\cdots\!34\)\( T^{16} + \)\(12\!\cdots\!72\)\( p^{4} T^{17} - \)\(12\!\cdots\!24\)\( p^{8} T^{18} - \)\(26\!\cdots\!16\)\( p^{12} T^{19} + \)\(20\!\cdots\!30\)\( p^{16} T^{20} + \)\(19\!\cdots\!08\)\( p^{20} T^{21} + \)\(44\!\cdots\!08\)\( p^{24} T^{22} + \)\(23\!\cdots\!60\)\( p^{28} T^{23} + \)\(84\!\cdots\!37\)\( p^{32} T^{24} - \)\(19\!\cdots\!48\)\( p^{36} T^{25} - \)\(23\!\cdots\!46\)\( p^{40} T^{26} - 284049247837485888 p^{44} T^{27} + 262036549366003 p^{48} T^{28} + 58882798008 p^{52} T^{29} - 17031278 p^{56} T^{30} - 4476 p^{60} T^{31} + p^{64} T^{32} \) | |
| 59 | \( 1 + 4668 T + 68987978 T^{2} + 288130292760 T^{3} + 2536595525664867 T^{4} + 9270034251586945248 T^{5} + \)\(60\!\cdots\!34\)\( T^{6} + \)\(18\!\cdots\!04\)\( T^{7} + \)\(99\!\cdots\!01\)\( T^{8} + \)\(22\!\cdots\!72\)\( T^{9} + \)\(10\!\cdots\!36\)\( T^{10} + \)\(76\!\cdots\!16\)\( T^{11} + \)\(34\!\cdots\!58\)\( T^{12} - \)\(31\!\cdots\!36\)\( T^{13} - \)\(91\!\cdots\!96\)\( T^{14} - \)\(76\!\cdots\!80\)\( T^{15} - \)\(18\!\cdots\!58\)\( T^{16} - \)\(76\!\cdots\!80\)\( p^{4} T^{17} - \)\(91\!\cdots\!96\)\( p^{8} T^{18} - \)\(31\!\cdots\!36\)\( p^{12} T^{19} + \)\(34\!\cdots\!58\)\( p^{16} T^{20} + \)\(76\!\cdots\!16\)\( p^{20} T^{21} + \)\(10\!\cdots\!36\)\( p^{24} T^{22} + \)\(22\!\cdots\!72\)\( p^{28} T^{23} + \)\(99\!\cdots\!01\)\( p^{32} T^{24} + \)\(18\!\cdots\!04\)\( p^{36} T^{25} + \)\(60\!\cdots\!34\)\( p^{40} T^{26} + 9270034251586945248 p^{44} T^{27} + 2536595525664867 p^{48} T^{28} + 288130292760 p^{52} T^{29} + 68987978 p^{56} T^{30} + 4668 p^{60} T^{31} + p^{64} T^{32} \) | |
| 61 | \( 1 - 17760 T + 210419048 T^{2} - 1869770100480 T^{3} + 13623437899930212 T^{4} - 84816490175649010464 T^{5} + \)\(46\!\cdots\!96\)\( T^{6} - \)\(37\!\cdots\!00\)\( p T^{7} + \)\(10\!\cdots\!10\)\( T^{8} - \)\(44\!\cdots\!72\)\( T^{9} + \)\(18\!\cdots\!84\)\( T^{10} - \)\(75\!\cdots\!56\)\( T^{11} + \)\(30\!\cdots\!52\)\( T^{12} - \)\(12\!\cdots\!68\)\( T^{13} + \)\(49\!\cdots\!68\)\( T^{14} - \)\(19\!\cdots\!96\)\( T^{15} + \)\(73\!\cdots\!71\)\( T^{16} - \)\(19\!\cdots\!96\)\( p^{4} T^{17} + \)\(49\!\cdots\!68\)\( p^{8} T^{18} - \)\(12\!\cdots\!68\)\( p^{12} T^{19} + \)\(30\!\cdots\!52\)\( p^{16} T^{20} - \)\(75\!\cdots\!56\)\( p^{20} T^{21} + \)\(18\!\cdots\!84\)\( p^{24} T^{22} - \)\(44\!\cdots\!72\)\( p^{28} T^{23} + \)\(10\!\cdots\!10\)\( p^{32} T^{24} - \)\(37\!\cdots\!00\)\( p^{37} T^{25} + \)\(46\!\cdots\!96\)\( p^{40} T^{26} - 84816490175649010464 p^{44} T^{27} + 13623437899930212 p^{48} T^{28} - 1869770100480 p^{52} T^{29} + 210419048 p^{56} T^{30} - 17760 p^{60} T^{31} + p^{64} T^{32} \) | |
| 67 | \( 1 - 1580 T - 99703854 T^{2} + 83109959736 T^{3} + 5067265037316531 T^{4} - 1229588707775966464 T^{5} - \)\(17\!\cdots\!70\)\( T^{6} - \)\(76\!\cdots\!44\)\( T^{7} + \)\(50\!\cdots\!01\)\( T^{8} + \)\(26\!\cdots\!68\)\( T^{9} - \)\(12\!\cdots\!56\)\( T^{10} + \)\(24\!\cdots\!44\)\( T^{11} + \)\(29\!\cdots\!22\)\( T^{12} - \)\(82\!\cdots\!12\)\( T^{13} - \)\(67\!\cdots\!28\)\( T^{14} + \)\(83\!\cdots\!32\)\( T^{15} + \)\(14\!\cdots\!22\)\( T^{16} + \)\(83\!\cdots\!32\)\( p^{4} T^{17} - \)\(67\!\cdots\!28\)\( p^{8} T^{18} - \)\(82\!\cdots\!12\)\( p^{12} T^{19} + \)\(29\!\cdots\!22\)\( p^{16} T^{20} + \)\(24\!\cdots\!44\)\( p^{20} T^{21} - \)\(12\!\cdots\!56\)\( p^{24} T^{22} + \)\(26\!\cdots\!68\)\( p^{28} T^{23} + \)\(50\!\cdots\!01\)\( p^{32} T^{24} - \)\(76\!\cdots\!44\)\( p^{36} T^{25} - \)\(17\!\cdots\!70\)\( p^{40} T^{26} - 1229588707775966464 p^{44} T^{27} + 5067265037316531 p^{48} T^{28} + 83109959736 p^{52} T^{29} - 99703854 p^{56} T^{30} - 1580 p^{60} T^{31} + p^{64} T^{32} \) | |
| 71 | \( ( 1 - 24 T + 112319632 T^{2} - 70727613192 T^{3} + 6191213832491356 T^{4} - 8795494210191006168 T^{5} + \)\(22\!\cdots\!44\)\( T^{6} - \)\(43\!\cdots\!08\)\( T^{7} + \)\(63\!\cdots\!42\)\( T^{8} - \)\(43\!\cdots\!08\)\( p^{4} T^{9} + \)\(22\!\cdots\!44\)\( p^{8} T^{10} - 8795494210191006168 p^{12} T^{11} + 6191213832491356 p^{16} T^{12} - 70727613192 p^{20} T^{13} + 112319632 p^{24} T^{14} - 24 p^{28} T^{15} + p^{32} T^{16} )^{2} \) | |
| 73 | \( 1 - 588 T + 131364762 T^{2} - 77174714232 T^{3} + 9043380460482531 T^{4} - 686585617896361728 T^{5} + \)\(39\!\cdots\!26\)\( T^{6} + \)\(29\!\cdots\!92\)\( T^{7} + \)\(11\!\cdots\!37\)\( T^{8} + \)\(24\!\cdots\!44\)\( T^{9} + \)\(23\!\cdots\!40\)\( T^{10} + \)\(10\!\cdots\!44\)\( T^{11} + \)\(24\!\cdots\!06\)\( T^{12} + \)\(30\!\cdots\!44\)\( T^{13} - \)\(14\!\cdots\!32\)\( T^{14} + \)\(75\!\cdots\!00\)\( T^{15} - \)\(12\!\cdots\!10\)\( T^{16} + \)\(75\!\cdots\!00\)\( p^{4} T^{17} - \)\(14\!\cdots\!32\)\( p^{8} T^{18} + \)\(30\!\cdots\!44\)\( p^{12} T^{19} + \)\(24\!\cdots\!06\)\( p^{16} T^{20} + \)\(10\!\cdots\!44\)\( p^{20} T^{21} + \)\(23\!\cdots\!40\)\( p^{24} T^{22} + \)\(24\!\cdots\!44\)\( p^{28} T^{23} + \)\(11\!\cdots\!37\)\( p^{32} T^{24} + \)\(29\!\cdots\!92\)\( p^{36} T^{25} + \)\(39\!\cdots\!26\)\( p^{40} T^{26} - 686585617896361728 p^{44} T^{27} + 9043380460482531 p^{48} T^{28} - 77174714232 p^{52} T^{29} + 131364762 p^{56} T^{30} - 588 p^{60} T^{31} + p^{64} T^{32} \) | |
| 79 | \( 1 + 3824 T - 206923908 T^{2} - 216241012800 T^{3} + 25547724077293590 T^{4} - 22024548599914585424 T^{5} - \)\(20\!\cdots\!04\)\( T^{6} + \)\(58\!\cdots\!60\)\( T^{7} + \)\(11\!\cdots\!73\)\( T^{8} - \)\(54\!\cdots\!80\)\( T^{9} - \)\(44\!\cdots\!72\)\( T^{10} + \)\(32\!\cdots\!92\)\( T^{11} + \)\(97\!\cdots\!18\)\( T^{12} - \)\(12\!\cdots\!60\)\( T^{13} + \)\(35\!\cdots\!44\)\( T^{14} + \)\(20\!\cdots\!56\)\( T^{15} - \)\(84\!\cdots\!84\)\( T^{16} + \)\(20\!\cdots\!56\)\( p^{4} T^{17} + \)\(35\!\cdots\!44\)\( p^{8} T^{18} - \)\(12\!\cdots\!60\)\( p^{12} T^{19} + \)\(97\!\cdots\!18\)\( p^{16} T^{20} + \)\(32\!\cdots\!92\)\( p^{20} T^{21} - \)\(44\!\cdots\!72\)\( p^{24} T^{22} - \)\(54\!\cdots\!80\)\( p^{28} T^{23} + \)\(11\!\cdots\!73\)\( p^{32} T^{24} + \)\(58\!\cdots\!60\)\( p^{36} T^{25} - \)\(20\!\cdots\!04\)\( p^{40} T^{26} - 22024548599914585424 p^{44} T^{27} + 25547724077293590 p^{48} T^{28} - 216241012800 p^{52} T^{29} - 206923908 p^{56} T^{30} + 3824 p^{60} T^{31} + p^{64} T^{32} \) | |
| 83 | \( 1 - 374548020 T^{2} + 72695408845030278 T^{4} - \)\(97\!\cdots\!08\)\( T^{6} + \)\(10\!\cdots\!13\)\( T^{8} - \)\(84\!\cdots\!32\)\( T^{10} + \)\(59\!\cdots\!38\)\( T^{12} - \)\(35\!\cdots\!88\)\( T^{14} + \)\(18\!\cdots\!92\)\( T^{16} - \)\(35\!\cdots\!88\)\( p^{8} T^{18} + \)\(59\!\cdots\!38\)\( p^{16} T^{20} - \)\(84\!\cdots\!32\)\( p^{24} T^{22} + \)\(10\!\cdots\!13\)\( p^{32} T^{24} - \)\(97\!\cdots\!08\)\( p^{40} T^{26} + 72695408845030278 p^{48} T^{28} - 374548020 p^{56} T^{30} + p^{64} T^{32} \) | |
| 89 | \( 1 + 360 T + 356618928 T^{2} + 128367262080 T^{3} + 69558682312465188 T^{4} + 15674686145039979480 T^{5} + \)\(90\!\cdots\!88\)\( T^{6} - \)\(41\!\cdots\!00\)\( T^{7} + \)\(84\!\cdots\!10\)\( T^{8} - \)\(46\!\cdots\!40\)\( T^{9} + \)\(58\!\cdots\!64\)\( T^{10} - \)\(86\!\cdots\!12\)\( T^{11} + \)\(30\!\cdots\!96\)\( T^{12} - \)\(10\!\cdots\!40\)\( T^{13} + \)\(12\!\cdots\!20\)\( T^{14} - \)\(84\!\cdots\!28\)\( T^{15} + \)\(57\!\cdots\!11\)\( T^{16} - \)\(84\!\cdots\!28\)\( p^{4} T^{17} + \)\(12\!\cdots\!20\)\( p^{8} T^{18} - \)\(10\!\cdots\!40\)\( p^{12} T^{19} + \)\(30\!\cdots\!96\)\( p^{16} T^{20} - \)\(86\!\cdots\!12\)\( p^{20} T^{21} + \)\(58\!\cdots\!64\)\( p^{24} T^{22} - \)\(46\!\cdots\!40\)\( p^{28} T^{23} + \)\(84\!\cdots\!10\)\( p^{32} T^{24} - \)\(41\!\cdots\!00\)\( p^{36} T^{25} + \)\(90\!\cdots\!88\)\( p^{40} T^{26} + 15674686145039979480 p^{44} T^{27} + 69558682312465188 p^{48} T^{28} + 128367262080 p^{52} T^{29} + 356618928 p^{56} T^{30} + 360 p^{60} T^{31} + p^{64} T^{32} \) | |
| 97 | \( 1 - 887851428 T^{2} + 399863674719063654 T^{4} - \)\(11\!\cdots\!88\)\( T^{6} + \)\(26\!\cdots\!17\)\( T^{8} - \)\(46\!\cdots\!80\)\( T^{10} + \)\(65\!\cdots\!98\)\( T^{12} - \)\(76\!\cdots\!88\)\( T^{14} + \)\(74\!\cdots\!64\)\( T^{16} - \)\(76\!\cdots\!88\)\( p^{8} T^{18} + \)\(65\!\cdots\!98\)\( p^{16} T^{20} - \)\(46\!\cdots\!80\)\( p^{24} T^{22} + \)\(26\!\cdots\!17\)\( p^{32} T^{24} - \)\(11\!\cdots\!88\)\( p^{40} T^{26} + 399863674719063654 p^{48} T^{28} - 887851428 p^{56} T^{30} + p^{64} T^{32} \) | |
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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
−2.79914241190998931366607112721, −2.73764190179469275634429238099, −2.70812438077730878226458194226, −2.59912056498356793368488544739, −2.47822717176569655041265974609, −2.08091212141361012522724442035, −2.02855222909271010474344586345, −1.92575908231232003563864467084, −1.90549496914284730039998311770, −1.88488061151090743019496087524, −1.82601848684543246667418138632, −1.72183375720038803063645114826, −1.52972773405898213625807738561, −1.30573585923869202268023153259, −1.12763433888258349345336013624, −1.11693741062495196629049120727, −1.02214943538238202633827823537, −0.892385152196412788346291630491, −0.865816096969303159344609202025, −0.50100356807770728416289305691, −0.45479499169504815764238808307, −0.43788530602779012103677409357, −0.31780344569406581140370555419, −0.06874215599536431937547602171, −0.05813843402144750011755659346, 0.05813843402144750011755659346, 0.06874215599536431937547602171, 0.31780344569406581140370555419, 0.43788530602779012103677409357, 0.45479499169504815764238808307, 0.50100356807770728416289305691, 0.865816096969303159344609202025, 0.892385152196412788346291630491, 1.02214943538238202633827823537, 1.11693741062495196629049120727, 1.12763433888258349345336013624, 1.30573585923869202268023153259, 1.52972773405898213625807738561, 1.72183375720038803063645114826, 1.82601848684543246667418138632, 1.88488061151090743019496087524, 1.90549496914284730039998311770, 1.92575908231232003563864467084, 2.02855222909271010474344586345, 2.08091212141361012522724442035, 2.47822717176569655041265974609, 2.59912056498356793368488544739, 2.70812438077730878226458194226, 2.73764190179469275634429238099, 2.79914241190998931366607112721
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.