| L(s) = 1 | − 4·2-s − 8.68·3-s + 16·4-s + 86.3·5-s + 34.7·6-s − 49·7-s − 64·8-s − 167.·9-s − 345.·10-s + 65.7·11-s − 138.·12-s − 169·13-s + 196·14-s − 749.·15-s + 256·16-s − 795.·17-s + 670.·18-s + 149.·19-s + 1.38e3·20-s + 425.·21-s − 262.·22-s + 267.·23-s + 555.·24-s + 4.32e3·25-s + 676·26-s + 3.56e3·27-s − 784·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.557·3-s + 0.5·4-s + 1.54·5-s + 0.393·6-s − 0.377·7-s − 0.353·8-s − 0.689·9-s − 1.09·10-s + 0.163·11-s − 0.278·12-s − 0.277·13-s + 0.267·14-s − 0.860·15-s + 0.250·16-s − 0.667·17-s + 0.487·18-s + 0.0952·19-s + 0.772·20-s + 0.210·21-s − 0.115·22-s + 0.105·23-s + 0.196·24-s + 1.38·25-s + 0.196·26-s + 0.941·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 7 | \( 1 + 49T \) |
| 13 | \( 1 + 169T \) |
| good | 3 | \( 1 + 8.68T + 243T^{2} \) |
| 5 | \( 1 - 86.3T + 3.12e3T^{2} \) |
| 11 | \( 1 - 65.7T + 1.61e5T^{2} \) |
| 17 | \( 1 + 795.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 149.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 267.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.06e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.90e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.80e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 7.88e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 8.45e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.59e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 7.04e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 8.85e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.48e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.40e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.54e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.57e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.53e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.02e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.05e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.39e5T + 8.58e9T^{2} \) |
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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
−11.04881571988698931575807515794, −10.13745544029961900508825429139, −9.388773854902870730071768470727, −8.449882949643541035388455927806, −6.80399866166411099829727952738, −6.10196468877018888189188658316, −5.09772407172162434222105151358, −2.88315567385033531637404971501, −1.62627093931810734613426382744, 0,
1.62627093931810734613426382744, 2.88315567385033531637404971501, 5.09772407172162434222105151358, 6.10196468877018888189188658316, 6.80399866166411099829727952738, 8.449882949643541035388455927806, 9.388773854902870730071768470727, 10.13745544029961900508825429139, 11.04881571988698931575807515794
