| L(s) = 1 | − 57·3-s − 500·5-s − 1.37e3·7-s + 322·9-s + 239·11-s − 7.33e3·13-s + 2.85e4·15-s − 1.51e4·17-s + 3.11e4·19-s + 7.82e4·21-s − 5.11e4·23-s + 1.56e5·25-s − 1.99e4·27-s − 4.14e4·29-s − 3.47e5·31-s − 1.36e4·33-s + 6.86e5·35-s + 1.96e5·37-s + 4.17e5·39-s + 1.08e6·41-s + 5.74e5·43-s − 1.61e5·45-s − 7.88e5·47-s + 1.17e6·49-s + 8.61e5·51-s + 1.63e6·53-s − 1.19e5·55-s + ⋯ |
| L(s) = 1 | − 1.21·3-s − 1.78·5-s − 1.51·7-s + 0.147·9-s + 0.0541·11-s − 0.925·13-s + 2.18·15-s − 0.745·17-s + 1.04·19-s + 1.84·21-s − 0.875·23-s + 2·25-s − 0.194·27-s − 0.315·29-s − 2.09·31-s − 0.0659·33-s + 2.70·35-s + 0.637·37-s + 1.12·39-s + 2.46·41-s + 1.10·43-s − 0.263·45-s − 1.10·47-s + 10/7·49-s + 0.909·51-s + 1.51·53-s − 0.0968·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p^{3} T )^{4} \) |
| 7 | $C_1$ | \( ( 1 + p^{3} T )^{4} \) |
| good | 3 | $C_2 \wr S_4$ | \( 1 + 19 p T + 2927 T^{2} + 56140 p T^{3} + 611600 p^{2} T^{4} + 56140 p^{8} T^{5} + 2927 p^{14} T^{6} + 19 p^{22} T^{7} + p^{28} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 239 T + 45391883 T^{2} - 8365752732 T^{3} + 1126066985219604 T^{4} - 8365752732 p^{7} T^{5} + 45391883 p^{14} T^{6} - 239 p^{21} T^{7} + p^{28} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 7331 T + 154813837 T^{2} + 460101585022 T^{3} + 9675867266709218 T^{4} + 460101585022 p^{7} T^{5} + 154813837 p^{14} T^{6} + 7331 p^{21} T^{7} + p^{28} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 15109 T + 1084735553 T^{2} + 14062304120910 T^{3} + 614596238200935510 T^{4} + 14062304120910 p^{7} T^{5} + 1084735553 p^{14} T^{6} + 15109 p^{21} T^{7} + p^{28} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 31174 T + 2838910744 T^{2} - 54393708797134 T^{3} + 3262876226039950366 T^{4} - 54393708797134 p^{7} T^{5} + 2838910744 p^{14} T^{6} - 31174 p^{21} T^{7} + p^{28} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 2222 p T + 3005074136 T^{2} + 242678854069890 T^{3} + 23562373345059287022 T^{4} + 242678854069890 p^{7} T^{5} + 3005074136 p^{14} T^{6} + 2222 p^{22} T^{7} + p^{28} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 41493 T + 13135949117 T^{2} + 2649499320115458 T^{3} - 88849499066320909998 T^{4} + 2649499320115458 p^{7} T^{5} + 13135949117 p^{14} T^{6} + 41493 p^{21} T^{7} + p^{28} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 347764 T + 93326654428 T^{2} + 20628478386343076 T^{3} + \)\(41\!\cdots\!82\)\( T^{4} + 20628478386343076 p^{7} T^{5} + 93326654428 p^{14} T^{6} + 347764 p^{21} T^{7} + p^{28} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 196300 T + 204230223316 T^{2} - 54926011233752420 T^{3} + \)\(22\!\cdots\!78\)\( T^{4} - 54926011233752420 p^{7} T^{5} + 204230223316 p^{14} T^{6} - 196300 p^{21} T^{7} + p^{28} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 1088846 T + 1116215110748 T^{2} - 662396410424203482 T^{3} + \)\(35\!\cdots\!54\)\( T^{4} - 662396410424203482 p^{7} T^{5} + 1116215110748 p^{14} T^{6} - 1088846 p^{21} T^{7} + p^{28} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 574638 T + 1002063478192 T^{2} - 405265244471039590 T^{3} + \)\(39\!\cdots\!10\)\( T^{4} - 405265244471039590 p^{7} T^{5} + 1002063478192 p^{14} T^{6} - 574638 p^{21} T^{7} + p^{28} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 788941 T + 918758847095 T^{2} + 67455056199399288 T^{3} + \)\(17\!\cdots\!88\)\( T^{4} + 67455056199399288 p^{7} T^{5} + 918758847095 p^{14} T^{6} + 788941 p^{21} T^{7} + p^{28} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 1638714 T + 5187153478172 T^{2} - 5599064725388791254 T^{3} + \)\(93\!\cdots\!90\)\( T^{4} - 5599064725388791254 p^{7} T^{5} + 5187153478172 p^{14} T^{6} - 1638714 p^{21} T^{7} + p^{28} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 112480 T + 6777865124684 T^{2} + 2932135482450355296 T^{3} + \)\(21\!\cdots\!10\)\( T^{4} + 2932135482450355296 p^{7} T^{5} + 6777865124684 p^{14} T^{6} + 112480 p^{21} T^{7} + p^{28} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 5086354 T + 20813049673300 T^{2} - 52633392555255256526 T^{3} + \)\(11\!\cdots\!58\)\( T^{4} - 52633392555255256526 p^{7} T^{5} + 20813049673300 p^{14} T^{6} - 5086354 p^{21} T^{7} + p^{28} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 3751752 T + 12111078198700 T^{2} + 8929914956099757512 T^{3} + \)\(17\!\cdots\!82\)\( T^{4} + 8929914956099757512 p^{7} T^{5} + 12111078198700 p^{14} T^{6} + 3751752 p^{21} T^{7} + p^{28} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 3394240 T + 1060305211100 T^{2} - 16789159683018018240 T^{3} + \)\(13\!\cdots\!38\)\( T^{4} - 16789159683018018240 p^{7} T^{5} + 1060305211100 p^{14} T^{6} - 3394240 p^{21} T^{7} + p^{28} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 8473576 T + 64700534403964 T^{2} - \)\(29\!\cdots\!52\)\( T^{3} + \)\(11\!\cdots\!98\)\( T^{4} - \)\(29\!\cdots\!52\)\( p^{7} T^{5} + 64700534403964 p^{14} T^{6} - 8473576 p^{21} T^{7} + p^{28} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 13011243 T + 129152375559751 T^{2} + \)\(81\!\cdots\!52\)\( T^{3} + \)\(42\!\cdots\!56\)\( T^{4} + \)\(81\!\cdots\!52\)\( p^{7} T^{5} + 129152375559751 p^{14} T^{6} + 13011243 p^{21} T^{7} + p^{28} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 10175868 T + 116154459355292 T^{2} + \)\(77\!\cdots\!20\)\( T^{3} + \)\(47\!\cdots\!70\)\( T^{4} + \)\(77\!\cdots\!20\)\( p^{7} T^{5} + 116154459355292 p^{14} T^{6} + 10175868 p^{21} T^{7} + p^{28} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 3861206 T + 37826456877692 T^{2} - \)\(37\!\cdots\!38\)\( T^{3} + \)\(33\!\cdots\!94\)\( T^{4} - \)\(37\!\cdots\!38\)\( p^{7} T^{5} + 37826456877692 p^{14} T^{6} - 3861206 p^{21} T^{7} + p^{28} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 12012703 T + 167555789176177 T^{2} - \)\(15\!\cdots\!54\)\( T^{3} + \)\(19\!\cdots\!98\)\( T^{4} - \)\(15\!\cdots\!54\)\( p^{7} T^{5} + 167555789176177 p^{14} T^{6} - 12012703 p^{21} T^{7} + p^{28} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.25881562713657709820936080353, −6.92199641552287426690210613066, −6.51448798273479195257855026299, −6.33898669452087254179998273102, −6.18339515764635414985050010538, −5.87469568423980257486195503885, −5.45693314813358665274448393333, −5.44935060129951207618966352188, −5.43941873716047765740798865374, −4.87113002204801221275345641283, −4.59179811021244250791098481609, −4.40603339256954049332754144904, −4.05679399356584730405040403185, −3.78882965717350662000546365931, −3.72565906007633380611027274930, −3.53560305256351968703936569232, −3.16940193727342868183402602865, −2.73564695979609770708331307724, −2.47994666536500017551328266134, −2.32278254640090436497369688341, −2.17000224888032096274224019979, −1.47486404136886046176992481758, −1.06183465314298214042388404557, −0.928135273838899565178278103930, −0.70049961948184591788650772008, 0, 0, 0, 0,
0.70049961948184591788650772008, 0.928135273838899565178278103930, 1.06183465314298214042388404557, 1.47486404136886046176992481758, 2.17000224888032096274224019979, 2.32278254640090436497369688341, 2.47994666536500017551328266134, 2.73564695979609770708331307724, 3.16940193727342868183402602865, 3.53560305256351968703936569232, 3.72565906007633380611027274930, 3.78882965717350662000546365931, 4.05679399356584730405040403185, 4.40603339256954049332754144904, 4.59179811021244250791098481609, 4.87113002204801221275345641283, 5.43941873716047765740798865374, 5.44935060129951207618966352188, 5.45693314813358665274448393333, 5.87469568423980257486195503885, 6.18339515764635414985050010538, 6.33898669452087254179998273102, 6.51448798273479195257855026299, 6.92199641552287426690210613066, 7.25881562713657709820936080353
