| L(s) = 1 | + 32·2-s + 1.02e3·4-s − 7.35e3·5-s + 1.68e4·7-s + 3.27e4·8-s − 2.35e5·10-s + 1.08e5·11-s − 6.35e5·13-s + 5.37e5·14-s + 1.04e6·16-s + 9.22e6·17-s − 7.55e6·19-s − 7.52e6·20-s + 3.48e6·22-s − 2.64e7·23-s + 5.19e6·25-s − 2.03e7·26-s + 1.72e7·28-s + 1.69e8·29-s − 5.13e7·31-s + 3.35e7·32-s + 2.95e8·34-s − 1.23e8·35-s − 2.51e8·37-s − 2.41e8·38-s − 2.40e8·40-s + 9.28e8·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.05·5-s + 0.377·7-s + 0.353·8-s − 0.743·10-s + 0.203·11-s − 0.474·13-s + 0.267·14-s + 1/4·16-s + 1.57·17-s − 0.700·19-s − 0.525·20-s + 0.144·22-s − 0.858·23-s + 0.106·25-s − 0.335·26-s + 0.188·28-s + 1.53·29-s − 0.322·31-s + 0.176·32-s + 1.11·34-s − 0.397·35-s − 0.596·37-s − 0.494·38-s − 0.371·40-s + 1.25·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{5} T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - p^{5} T \) |
| good | 5 | \( 1 + 294 p^{2} T + p^{11} T^{2} \) |
| 11 | \( 1 - 108780 T + p^{11} T^{2} \) |
| 13 | \( 1 + 635842 T + p^{11} T^{2} \) |
| 17 | \( 1 - 9225918 T + p^{11} T^{2} \) |
| 19 | \( 1 + 7555372 T + p^{11} T^{2} \) |
| 23 | \( 1 + 26489400 T + p^{11} T^{2} \) |
| 29 | \( 1 - 169827594 T + p^{11} T^{2} \) |
| 31 | \( 1 + 51362704 T + p^{11} T^{2} \) |
| 37 | \( 1 + 251605906 T + p^{11} T^{2} \) |
| 41 | \( 1 - 928817814 T + p^{11} T^{2} \) |
| 43 | \( 1 + 1818895756 T + p^{11} T^{2} \) |
| 47 | \( 1 + 523343136 T + p^{11} T^{2} \) |
| 53 | \( 1 + 4199520078 T + p^{11} T^{2} \) |
| 59 | \( 1 + 154917444 p T + p^{11} T^{2} \) |
| 61 | \( 1 + 6639312802 T + p^{11} T^{2} \) |
| 67 | \( 1 + 2878139188 T + p^{11} T^{2} \) |
| 71 | \( 1 - 4345596360 T + p^{11} T^{2} \) |
| 73 | \( 1 - 23450332826 T + p^{11} T^{2} \) |
| 79 | \( 1 + 28761853648 T + p^{11} T^{2} \) |
| 83 | \( 1 - 5577757548 T + p^{11} T^{2} \) |
| 89 | \( 1 + 78002173386 T + p^{11} T^{2} \) |
| 97 | \( 1 + 26685859630 T + p^{11} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.98455119994997746576383094170, −9.886958893114514737848924241537, −8.262492185386106091642464214988, −7.57925763456180521732364816778, −6.31391413393239773586483703061, −5.00740246005165269892785988001, −4.03944325605094753928260174265, −3.00546159702155120106801852146, −1.48549702440249427632711579043, 0,
1.48549702440249427632711579043, 3.00546159702155120106801852146, 4.03944325605094753928260174265, 5.00740246005165269892785988001, 6.31391413393239773586483703061, 7.57925763456180521732364816778, 8.262492185386106091642464214988, 9.886958893114514737848924241537, 10.98455119994997746576383094170
