| L(s) = 1 | − 2-s − 3-s + 4-s − 3.97·5-s + 6-s + 2.10·7-s − 8-s + 9-s + 3.97·10-s − 2.46·11-s − 12-s − 3.30·13-s − 2.10·14-s + 3.97·15-s + 16-s − 1.57·17-s − 18-s − 19-s − 3.97·20-s − 2.10·21-s + 2.46·22-s + 5.45·23-s + 24-s + 10.7·25-s + 3.30·26-s − 27-s + 2.10·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.77·5-s + 0.408·6-s + 0.797·7-s − 0.353·8-s + 0.333·9-s + 1.25·10-s − 0.741·11-s − 0.288·12-s − 0.916·13-s − 0.563·14-s + 1.02·15-s + 0.250·16-s − 0.381·17-s − 0.235·18-s − 0.229·19-s − 0.888·20-s − 0.460·21-s + 0.524·22-s + 1.13·23-s + 0.204·24-s + 2.15·25-s + 0.647·26-s − 0.192·27-s + 0.398·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 + T \) |
| good | 5 | \( 1 + 3.97T + 5T^{2} \) |
| 7 | \( 1 - 2.10T + 7T^{2} \) |
| 11 | \( 1 + 2.46T + 11T^{2} \) |
| 13 | \( 1 + 3.30T + 13T^{2} \) |
| 17 | \( 1 + 1.57T + 17T^{2} \) |
| 23 | \( 1 - 5.45T + 23T^{2} \) |
| 29 | \( 1 + 10.0T + 29T^{2} \) |
| 31 | \( 1 - 5.53T + 31T^{2} \) |
| 37 | \( 1 - 9.63T + 37T^{2} \) |
| 41 | \( 1 + 1.80T + 41T^{2} \) |
| 43 | \( 1 - 2.92T + 43T^{2} \) |
| 47 | \( 1 - 3.51T + 47T^{2} \) |
| 59 | \( 1 - 2.26T + 59T^{2} \) |
| 61 | \( 1 + 0.865T + 61T^{2} \) |
| 67 | \( 1 - 5.26T + 67T^{2} \) |
| 71 | \( 1 + 0.404T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 - 8.91T + 79T^{2} \) |
| 83 | \( 1 - 2.47T + 83T^{2} \) |
| 89 | \( 1 - 13.5T + 89T^{2} \) |
| 97 | \( 1 + 7.25T + 97T^{2} \) |
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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
−7.67441556335302894322477205323, −7.35674496274830618612062107165, −6.57809676125529651711149123020, −5.47676388710629850451244132526, −4.75427889448656488530382336160, −4.21586096550890250567729108915, −3.16981738331489980859382461074, −2.23740556503985550547882174533, −0.899670331681225878305189241440, 0,
0.899670331681225878305189241440, 2.23740556503985550547882174533, 3.16981738331489980859382461074, 4.21586096550890250567729108915, 4.75427889448656488530382336160, 5.47676388710629850451244132526, 6.57809676125529651711149123020, 7.35674496274830618612062107165, 7.67441556335302894322477205323
