| L(s) = 1 | + 0.569·2-s + 3-s − 1.67·4-s + 1.84·5-s + 0.569·6-s + 1.78·7-s − 2.09·8-s + 9-s + 1.04·10-s − 4.77·11-s − 1.67·12-s + 13-s + 1.01·14-s + 1.84·15-s + 2.16·16-s − 3.17·17-s + 0.569·18-s − 4.39·19-s − 3.08·20-s + 1.78·21-s − 2.72·22-s + 1.47·23-s − 2.09·24-s − 1.60·25-s + 0.569·26-s + 27-s − 2.99·28-s + ⋯ |
| L(s) = 1 | + 0.402·2-s + 0.577·3-s − 0.838·4-s + 0.824·5-s + 0.232·6-s + 0.676·7-s − 0.739·8-s + 0.333·9-s + 0.331·10-s − 1.44·11-s − 0.483·12-s + 0.277·13-s + 0.272·14-s + 0.475·15-s + 0.540·16-s − 0.768·17-s + 0.134·18-s − 1.00·19-s − 0.690·20-s + 0.390·21-s − 0.580·22-s + 0.306·23-s − 0.427·24-s − 0.320·25-s + 0.111·26-s + 0.192·27-s − 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 - 0.569T + 2T^{2} \) |
| 5 | \( 1 - 1.84T + 5T^{2} \) |
| 7 | \( 1 - 1.78T + 7T^{2} \) |
| 11 | \( 1 + 4.77T + 11T^{2} \) |
| 17 | \( 1 + 3.17T + 17T^{2} \) |
| 19 | \( 1 + 4.39T + 19T^{2} \) |
| 23 | \( 1 - 1.47T + 23T^{2} \) |
| 29 | \( 1 + 8.18T + 29T^{2} \) |
| 31 | \( 1 + 1.20T + 31T^{2} \) |
| 37 | \( 1 - 0.855T + 37T^{2} \) |
| 41 | \( 1 - 11.2T + 41T^{2} \) |
| 43 | \( 1 + 6.08T + 43T^{2} \) |
| 47 | \( 1 + 7.31T + 47T^{2} \) |
| 53 | \( 1 + 13.8T + 53T^{2} \) |
| 59 | \( 1 - 13.2T + 59T^{2} \) |
| 61 | \( 1 + 10.2T + 61T^{2} \) |
| 67 | \( 1 + 2.32T + 67T^{2} \) |
| 71 | \( 1 - 0.785T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 - 8.87T + 79T^{2} \) |
| 83 | \( 1 + 0.134T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 + 9.39T + 97T^{2} \) |
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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
−8.100056791075091080442423262290, −7.62027325195411945468017321606, −6.40702846430819791038395601702, −5.71260964151242855776707967246, −4.98058612791302401174882499708, −4.40450726802450558164617388585, −3.44192839636867539524534565292, −2.47249018810347952202041840440, −1.70594273614145005569111862205, 0,
1.70594273614145005569111862205, 2.47249018810347952202041840440, 3.44192839636867539524534565292, 4.40450726802450558164617388585, 4.98058612791302401174882499708, 5.71260964151242855776707967246, 6.40702846430819791038395601702, 7.62027325195411945468017321606, 8.100056791075091080442423262290
