| L(s) = 1 | + 2-s + 4-s − 5-s − 4.32i·7-s + 8-s − 10-s + 2.34·11-s − 1.88i·13-s − 4.32i·14-s + 16-s − 4.76i·17-s − 7.45·19-s − 20-s + 2.34·22-s + 0.306i·23-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.447·5-s − 1.63i·7-s + 0.353·8-s − 0.316·10-s + 0.706·11-s − 0.523i·13-s − 1.15i·14-s + 0.250·16-s − 1.15i·17-s − 1.70·19-s − 0.223·20-s + 0.499·22-s + 0.0638i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.445i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.895 + 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.814453802\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.814453802\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (7.20 + 3.87i)T \) |
| good | 7 | \( 1 + 4.32iT - 7T^{2} \) |
| 11 | \( 1 - 2.34T + 11T^{2} \) |
| 13 | \( 1 + 1.88iT - 13T^{2} \) |
| 17 | \( 1 + 4.76iT - 17T^{2} \) |
| 19 | \( 1 + 7.45T + 19T^{2} \) |
| 23 | \( 1 - 0.306iT - 23T^{2} \) |
| 29 | \( 1 - 1.02iT - 29T^{2} \) |
| 31 | \( 1 - 3.24iT - 31T^{2} \) |
| 37 | \( 1 - 4.46T + 37T^{2} \) |
| 41 | \( 1 + 1.04T + 41T^{2} \) |
| 43 | \( 1 + 7.21iT - 43T^{2} \) |
| 47 | \( 1 + 8.58iT - 47T^{2} \) |
| 53 | \( 1 + 0.890T + 53T^{2} \) |
| 59 | \( 1 - 2.70iT - 59T^{2} \) |
| 61 | \( 1 - 8.48iT - 61T^{2} \) |
| 71 | \( 1 - 7.06iT - 71T^{2} \) |
| 73 | \( 1 + 8.74T + 73T^{2} \) |
| 79 | \( 1 - 10.0iT - 79T^{2} \) |
| 83 | \( 1 + 3.81iT - 83T^{2} \) |
| 89 | \( 1 + 7.38iT - 89T^{2} \) |
| 97 | \( 1 - 11.2iT - 97T^{2} \) |
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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
−7.51821821797388716081880320307, −7.00518204219919588243551387362, −6.53787522800168875583034430589, −5.54144965659992104402514660754, −4.63271458218467557923533379355, −4.13085133737591850760982266331, −3.54823683206851131649257987937, −2.60603821420036504555260435584, −1.36802585401607003355168178834, −0.34061180276320126310089368173,
1.61945297815794657240155188355, 2.32331483275660569996234838565, 3.18555014292129966790411550979, 4.18576746060554594749396330411, 4.55901980896748222605013132286, 5.64562264159583663510129843497, 6.28849326908706682610521174580, 6.52851750129745196677646945576, 7.78207175589316372270267808076, 8.323921772509750594660351331873
