| L(s) = 1 | + 2-s − 0.950·3-s + 4-s − 2.70·5-s − 0.950·6-s + 7-s + 8-s − 2.09·9-s − 2.70·10-s + 1.16·11-s − 0.950·12-s − 6.37·13-s + 14-s + 2.57·15-s + 16-s + 5.68·17-s − 2.09·18-s + 3.93·19-s − 2.70·20-s − 0.950·21-s + 1.16·22-s + 2.75·23-s − 0.950·24-s + 2.33·25-s − 6.37·26-s + 4.84·27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.549·3-s + 0.5·4-s − 1.21·5-s − 0.388·6-s + 0.377·7-s + 0.353·8-s − 0.698·9-s − 0.856·10-s + 0.352·11-s − 0.274·12-s − 1.76·13-s + 0.267·14-s + 0.665·15-s + 0.250·16-s + 1.37·17-s − 0.493·18-s + 0.903·19-s − 0.605·20-s − 0.207·21-s + 0.248·22-s + 0.575·23-s − 0.194·24-s + 0.467·25-s − 1.25·26-s + 0.932·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 + 0.950T + 3T^{2} \) |
| 5 | \( 1 + 2.70T + 5T^{2} \) |
| 11 | \( 1 - 1.16T + 11T^{2} \) |
| 13 | \( 1 + 6.37T + 13T^{2} \) |
| 17 | \( 1 - 5.68T + 17T^{2} \) |
| 19 | \( 1 - 3.93T + 19T^{2} \) |
| 23 | \( 1 - 2.75T + 23T^{2} \) |
| 29 | \( 1 + 8.04T + 29T^{2} \) |
| 31 | \( 1 - 7.39T + 31T^{2} \) |
| 37 | \( 1 - 9.23T + 37T^{2} \) |
| 41 | \( 1 - 3.34T + 41T^{2} \) |
| 43 | \( 1 + 9.44T + 43T^{2} \) |
| 47 | \( 1 - 0.544T + 47T^{2} \) |
| 53 | \( 1 + 2.70T + 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 - 4.95T + 61T^{2} \) |
| 67 | \( 1 + 10.0T + 67T^{2} \) |
| 71 | \( 1 + 9.18T + 71T^{2} \) |
| 73 | \( 1 + 2.53T + 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 - 10.6T + 83T^{2} \) |
| 89 | \( 1 - 5.37T + 89T^{2} \) |
| 97 | \( 1 - 0.731T + 97T^{2} \) |
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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
−7.73198523875603104294054259490, −7.09768730656259621775612810154, −6.15859766554262553498894094406, −5.39136135138555379830503548636, −4.88418944828004827911056659542, −4.18790515241028805281005375828, −3.23719633559307191625986678300, −2.68437085154685544635400329528, −1.22206800807636150046042556694, 0,
1.22206800807636150046042556694, 2.68437085154685544635400329528, 3.23719633559307191625986678300, 4.18790515241028805281005375828, 4.88418944828004827911056659542, 5.39136135138555379830503548636, 6.15859766554262553498894094406, 7.09768730656259621775612810154, 7.73198523875603104294054259490
