| L(s) = 1 | + 3·5-s − 4·7-s + 7·11-s + 3·13-s − 6·17-s + 8·19-s + 2·23-s + 12·25-s + 9·29-s − 3·31-s − 12·35-s + 6·37-s − 9·41-s − 8·43-s + 3·47-s + 6·49-s − 12·53-s + 21·55-s + 10·59-s + 20·61-s + 9·65-s − 11·67-s − 6·71-s − 48·73-s − 28·77-s − 21·79-s + 8·83-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 1.51·7-s + 2.11·11-s + 0.832·13-s − 1.45·17-s + 1.83·19-s + 0.417·23-s + 12/5·25-s + 1.67·29-s − 0.538·31-s − 2.02·35-s + 0.986·37-s − 1.40·41-s − 1.21·43-s + 0.437·47-s + 6/7·49-s − 1.64·53-s + 2.83·55-s + 1.30·59-s + 2.56·61-s + 1.11·65-s − 1.34·67-s − 0.712·71-s − 5.61·73-s − 3.19·77-s − 2.36·79-s + 0.878·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.424779447\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.424779447\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( ( 1 + T + T^{2} )^{4} \) |
| good | 5 | \( 1 - 3 T - 3 T^{2} + 36 T^{3} - 32 T^{4} - 204 T^{5} + 108 p T^{6} + 597 T^{7} - 3831 T^{8} + 597 p T^{9} + 108 p^{3} T^{10} - 204 p^{3} T^{11} - 32 p^{4} T^{12} + 36 p^{5} T^{13} - 3 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) |
| 11 | \( 1 - 7 T - 2 T^{2} + 81 T^{3} + 203 T^{4} - 1468 T^{5} - 675 T^{6} + 3769 T^{7} + 21730 T^{8} + 3769 p T^{9} - 675 p^{2} T^{10} - 1468 p^{3} T^{11} + 203 p^{4} T^{12} + 81 p^{5} T^{13} - 2 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \) |
| 13 | \( 1 - 3 T - 32 T^{2} + 57 T^{3} + 673 T^{4} - 432 T^{5} - 11810 T^{6} + 2142 T^{7} + 166240 T^{8} + 2142 p T^{9} - 11810 p^{2} T^{10} - 432 p^{3} T^{11} + 673 p^{4} T^{12} + 57 p^{5} T^{13} - 32 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) |
| 17 | \( ( 1 + 3 T + 41 T^{2} + 189 T^{3} + 807 T^{4} + 189 p T^{5} + 41 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 4 T + 49 T^{2} - 193 T^{3} + 1297 T^{4} - 193 p T^{5} + 49 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( 1 - 2 T - 59 T^{2} + 192 T^{3} + 1775 T^{4} - 6170 T^{5} - 33567 T^{6} + 71510 T^{7} + 655807 T^{8} + 71510 p T^{9} - 33567 p^{2} T^{10} - 6170 p^{3} T^{11} + 1775 p^{4} T^{12} + 192 p^{5} T^{13} - 59 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 29 | \( 1 - 9 T + 13 T^{2} - 72 T^{3} + 904 T^{4} - 2970 T^{5} + 29086 T^{6} - 17613 T^{7} - 1095047 T^{8} - 17613 p T^{9} + 29086 p^{2} T^{10} - 2970 p^{3} T^{11} + 904 p^{4} T^{12} - 72 p^{5} T^{13} + 13 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) |
| 31 | \( 1 + 3 T - 41 T^{2} - 354 T^{3} - 128 T^{4} + 9180 T^{5} + 42568 T^{6} - 75933 T^{7} - 1254023 T^{8} - 75933 p T^{9} + 42568 p^{2} T^{10} + 9180 p^{3} T^{11} - 128 p^{4} T^{12} - 354 p^{5} T^{13} - 41 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) |
| 37 | \( ( 1 - 3 T + 140 T^{2} - 324 T^{3} + 7620 T^{4} - 324 p T^{5} + 140 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( 1 + 9 T - 36 T^{2} - 381 T^{3} + 895 T^{4} - 852 T^{5} - 99027 T^{6} + 57981 T^{7} + 4458864 T^{8} + 57981 p T^{9} - 99027 p^{2} T^{10} - 852 p^{3} T^{11} + 895 p^{4} T^{12} - 381 p^{5} T^{13} - 36 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \) |
| 43 | \( 1 + 8 T - 73 T^{2} - 914 T^{3} + 2642 T^{4} + 51508 T^{5} + 63012 T^{6} - 1050915 T^{7} - 5977292 T^{8} - 1050915 p T^{9} + 63012 p^{2} T^{10} + 51508 p^{3} T^{11} + 2642 p^{4} T^{12} - 914 p^{5} T^{13} - 73 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) |
| 47 | \( 1 - 3 T - 3 p T^{2} + 486 T^{3} + 11422 T^{4} - 34152 T^{5} - 635364 T^{6} + 765849 T^{7} + 30578781 T^{8} + 765849 p T^{9} - 635364 p^{2} T^{10} - 34152 p^{3} T^{11} + 11422 p^{4} T^{12} + 486 p^{5} T^{13} - 3 p^{7} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) |
| 53 | \( ( 1 + 6 T + 59 T^{2} + 621 T^{3} + 4704 T^{4} + 621 p T^{5} + 59 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 59 | \( 1 - 10 T - 71 T^{2} + 1536 T^{3} - 346 T^{4} - 106150 T^{5} + 583014 T^{6} + 3064183 T^{7} - 49833644 T^{8} + 3064183 p T^{9} + 583014 p^{2} T^{10} - 106150 p^{3} T^{11} - 346 p^{4} T^{12} + 1536 p^{5} T^{13} - 71 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) |
| 61 | \( 1 - 20 T + 161 T^{2} - 562 T^{3} - 931 T^{4} + 2180 T^{5} - 139869 T^{6} + 5694174 T^{7} - 64208639 T^{8} + 5694174 p T^{9} - 139869 p^{2} T^{10} + 2180 p^{3} T^{11} - 931 p^{4} T^{12} - 562 p^{5} T^{13} + 161 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \) |
| 67 | \( 1 + 11 T - 82 T^{2} - 689 T^{3} + 8039 T^{4} + 18292 T^{5} - 590799 T^{6} - 1428153 T^{7} + 17009218 T^{8} - 1428153 p T^{9} - 590799 p^{2} T^{10} + 18292 p^{3} T^{11} + 8039 p^{4} T^{12} - 689 p^{5} T^{13} - 82 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \) |
| 71 | \( ( 1 + 3 T + 276 T^{2} + 630 T^{3} + 29126 T^{4} + 630 p T^{5} + 276 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 24 T + 425 T^{2} + 4923 T^{3} + 48495 T^{4} + 4923 p T^{5} + 425 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( 1 + 21 T + 133 T^{2} - 252 T^{3} - 9896 T^{4} - 136080 T^{5} - 973784 T^{6} + 2121567 T^{7} + 81625693 T^{8} + 2121567 p T^{9} - 973784 p^{2} T^{10} - 136080 p^{3} T^{11} - 9896 p^{4} T^{12} - 252 p^{5} T^{13} + 133 p^{6} T^{14} + 21 p^{7} T^{15} + p^{8} T^{16} \) |
| 83 | \( 1 - 8 T - 259 T^{2} + 1146 T^{3} + 49577 T^{4} - 119878 T^{5} - 6175577 T^{6} + 3335418 T^{7} + 604903483 T^{8} + 3335418 p T^{9} - 6175577 p^{2} T^{10} - 119878 p^{3} T^{11} + 49577 p^{4} T^{12} + 1146 p^{5} T^{13} - 259 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) |
| 89 | \( ( 1 + 6 T + 263 T^{2} + 1377 T^{3} + 32646 T^{4} + 1377 p T^{5} + 263 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 97 | \( 1 - 16 T - 175 T^{2} + 2086 T^{3} + 49490 T^{4} - 309968 T^{5} - 6453258 T^{6} + 6735477 T^{7} + 823825414 T^{8} + 6735477 p T^{9} - 6453258 p^{2} T^{10} - 309968 p^{3} T^{11} + 49490 p^{4} T^{12} + 2086 p^{5} T^{13} - 175 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.72349941412404398185929403916, −3.32144030920750393490579773172, −3.31341875190398399814795481868, −3.31011052240973908170503416909, −3.24169903761344562155933634829, −2.87439146887436528678278164299, −2.82596076244788986177645948345, −2.82127141685893433827236661382, −2.64564698158596195657462916872, −2.61172935317712770268397436363, −2.57755435228216145759356435070, −2.41874910870923766428352748612, −2.12201334900233135896083199526, −2.07112994220007601852262094043, −1.67066312180721178375582406041, −1.57637790830242295760224577436, −1.51082273483431850333379143014, −1.41215723870430336305584032856, −1.29862791247995905687933834215, −1.22917870345209038998300422514, −1.07267093840228267993026118255, −1.00561266282661451801774813100, −0.58560841705053492336758568031, −0.19730834415011534258779813436, −0.14127717561670974242135606733,
0.14127717561670974242135606733, 0.19730834415011534258779813436, 0.58560841705053492336758568031, 1.00561266282661451801774813100, 1.07267093840228267993026118255, 1.22917870345209038998300422514, 1.29862791247995905687933834215, 1.41215723870430336305584032856, 1.51082273483431850333379143014, 1.57637790830242295760224577436, 1.67066312180721178375582406041, 2.07112994220007601852262094043, 2.12201334900233135896083199526, 2.41874910870923766428352748612, 2.57755435228216145759356435070, 2.61172935317712770268397436363, 2.64564698158596195657462916872, 2.82127141685893433827236661382, 2.82596076244788986177645948345, 2.87439146887436528678278164299, 3.24169903761344562155933634829, 3.31011052240973908170503416909, 3.31341875190398399814795481868, 3.32144030920750393490579773172, 3.72349941412404398185929403916
Plot not available for L-functions of degree greater than 10.
