| L(s) = 1 | − 1.47·2-s − 9.15·3-s − 5.82·4-s − 5·5-s + 13.4·6-s − 16.2·7-s + 20.3·8-s + 56.7·9-s + 7.36·10-s − 13.4·11-s + 53.3·12-s − 13·13-s + 24.0·14-s + 45.7·15-s + 16.6·16-s − 128.·17-s − 83.6·18-s − 57.6·19-s + 29.1·20-s + 149.·21-s + 19.7·22-s − 103.·23-s − 186.·24-s + 25·25-s + 19.1·26-s − 272.·27-s + 94.9·28-s + ⋯ |
| L(s) = 1 | − 0.520·2-s − 1.76·3-s − 0.728·4-s − 0.447·5-s + 0.917·6-s − 0.879·7-s + 0.900·8-s + 2.10·9-s + 0.232·10-s − 0.368·11-s + 1.28·12-s − 0.277·13-s + 0.458·14-s + 0.787·15-s + 0.259·16-s − 1.82·17-s − 1.09·18-s − 0.696·19-s + 0.325·20-s + 1.54·21-s + 0.191·22-s − 0.935·23-s − 1.58·24-s + 0.200·25-s + 0.144·26-s − 1.94·27-s + 0.640·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 13 | \( 1 + 13T \) |
| 31 | \( 1 - 31T \) |
| good | 2 | \( 1 + 1.47T + 8T^{2} \) |
| 3 | \( 1 + 9.15T + 27T^{2} \) |
| 7 | \( 1 + 16.2T + 343T^{2} \) |
| 11 | \( 1 + 13.4T + 1.33e3T^{2} \) |
| 17 | \( 1 + 128.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 57.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 103.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 31.6T + 2.43e4T^{2} \) |
| 37 | \( 1 + 111.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 18.8T + 6.89e4T^{2} \) |
| 43 | \( 1 + 410.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 479.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 20.7T + 1.48e5T^{2} \) |
| 59 | \( 1 + 235.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 451.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 125.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 248.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 458.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 889.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 503.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 504.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 122.T + 9.12e5T^{2} \) |
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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
−8.501022268261229065146835325937, −7.50516933480050754598823419165, −6.70317003597400527266417580002, −6.14119573130651484688962023642, −5.11516836347831338121653590256, −4.51434676087816684838249662034, −3.74489151505508219400271289278, −2.01417399359480384319768782455, −0.55325331516303581197494787915, 0,
0.55325331516303581197494787915, 2.01417399359480384319768782455, 3.74489151505508219400271289278, 4.51434676087816684838249662034, 5.11516836347831338121653590256, 6.14119573130651484688962023642, 6.70317003597400527266417580002, 7.50516933480050754598823419165, 8.501022268261229065146835325937
