| L(s) = 1 | − 5.14·2-s − 6.96·3-s + 18.5·4-s − 3.24·5-s + 35.8·6-s − 54.0·8-s + 21.4·9-s + 16.7·10-s + 58.8·11-s − 128.·12-s − 47.9·13-s + 22.5·15-s + 130.·16-s + 84.9·17-s − 110.·18-s − 54.8·19-s − 60.0·20-s − 302.·22-s + 150.·23-s + 376.·24-s − 114.·25-s + 246.·26-s + 38.6·27-s + 44.2·29-s − 116.·30-s − 95.7·31-s − 238.·32-s + ⋯ |
| L(s) = 1 | − 1.82·2-s − 1.33·3-s + 2.31·4-s − 0.290·5-s + 2.43·6-s − 2.38·8-s + 0.794·9-s + 0.528·10-s + 1.61·11-s − 3.09·12-s − 1.02·13-s + 0.389·15-s + 2.03·16-s + 1.21·17-s − 1.44·18-s − 0.662·19-s − 0.671·20-s − 2.93·22-s + 1.36·23-s + 3.19·24-s − 0.915·25-s + 1.86·26-s + 0.275·27-s + 0.283·29-s − 0.708·30-s − 0.554·31-s − 1.31·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| good | 2 | \( 1 + 5.14T + 8T^{2} \) |
| 3 | \( 1 + 6.96T + 27T^{2} \) |
| 5 | \( 1 + 3.24T + 125T^{2} \) |
| 11 | \( 1 - 58.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 47.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 84.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 54.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 150.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 44.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 95.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 209.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 354.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 221.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 235.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 508.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 439.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 51.4T + 2.26e5T^{2} \) |
| 67 | \( 1 + 263.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 203.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 679.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 189.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 520.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.49e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 712.T + 9.12e5T^{2} \) |
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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
−8.267229281042463833029792588323, −7.45725868310168630943340200517, −6.83344527151350419368327862906, −6.24892117113885936527527725292, −5.38807269294351406455495916253, −4.27158263607913548418963893814, −2.97625398661059338470275794082, −1.63426610561833189853844572359, −0.882509654731063653431930932475, 0,
0.882509654731063653431930932475, 1.63426610561833189853844572359, 2.97625398661059338470275794082, 4.27158263607913548418963893814, 5.38807269294351406455495916253, 6.24892117113885936527527725292, 6.83344527151350419368327862906, 7.45725868310168630943340200517, 8.267229281042463833029792588323
