| L(s) = 1 | + 2-s + 4-s − 5-s − 2.37·7-s + 8-s − 10-s + 2.84·11-s − 3.16·13-s − 2.37·14-s + 16-s + 6.06·17-s − 4.14·19-s − 20-s + 2.84·22-s − 4.98·23-s + 25-s − 3.16·26-s − 2.37·28-s − 7.76·29-s + 8.01·31-s + 32-s + 6.06·34-s + 2.37·35-s − 2.51·37-s − 4.14·38-s − 40-s + 4.84·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.447·5-s − 0.899·7-s + 0.353·8-s − 0.316·10-s + 0.858·11-s − 0.878·13-s − 0.636·14-s + 0.250·16-s + 1.46·17-s − 0.952·19-s − 0.223·20-s + 0.607·22-s − 1.03·23-s + 0.200·25-s − 0.621·26-s − 0.449·28-s − 1.44·29-s + 1.43·31-s + 0.176·32-s + 1.03·34-s + 0.402·35-s − 0.413·37-s − 0.673·38-s − 0.158·40-s + 0.757·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 + T \) |
| good | 7 | \( 1 + 2.37T + 7T^{2} \) |
| 11 | \( 1 - 2.84T + 11T^{2} \) |
| 13 | \( 1 + 3.16T + 13T^{2} \) |
| 17 | \( 1 - 6.06T + 17T^{2} \) |
| 19 | \( 1 + 4.14T + 19T^{2} \) |
| 23 | \( 1 + 4.98T + 23T^{2} \) |
| 29 | \( 1 + 7.76T + 29T^{2} \) |
| 31 | \( 1 - 8.01T + 31T^{2} \) |
| 37 | \( 1 + 2.51T + 37T^{2} \) |
| 41 | \( 1 - 4.84T + 41T^{2} \) |
| 43 | \( 1 - 8.90T + 43T^{2} \) |
| 47 | \( 1 + 6.37T + 47T^{2} \) |
| 53 | \( 1 - 8.75T + 53T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 - 1.83T + 61T^{2} \) |
| 71 | \( 1 + 14.0T + 71T^{2} \) |
| 73 | \( 1 + 16.9T + 73T^{2} \) |
| 79 | \( 1 + 10.6T + 79T^{2} \) |
| 83 | \( 1 + 14.2T + 83T^{2} \) |
| 89 | \( 1 + 11.6T + 89T^{2} \) |
| 97 | \( 1 - 2.57T + 97T^{2} \) |
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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
−7.46449496609139438473981483167, −7.07833596586379149963301701757, −5.98724717411462830595007917113, −5.88537713465234367933364571185, −4.60523724137589320429139376748, −4.08432432135515941184848032019, −3.32045469675287679981659229194, −2.59399703226325534454288461765, −1.43444921983442575613122225837, 0,
1.43444921983442575613122225837, 2.59399703226325534454288461765, 3.32045469675287679981659229194, 4.08432432135515941184848032019, 4.60523724137589320429139376748, 5.88537713465234367933364571185, 5.98724717411462830595007917113, 7.07833596586379149963301701757, 7.46449496609139438473981483167
