| L(s) = 1 | + 2.50·2-s + 4.25·4-s + 5-s − 4.89·7-s + 5.63·8-s + 2.50·10-s − 5.13·11-s + 0.293·13-s − 12.2·14-s + 5.58·16-s − 4.78·17-s + 4.79·19-s + 4.25·20-s − 12.8·22-s − 5.18·23-s + 25-s + 0.734·26-s − 20.8·28-s − 8.87·29-s − 6.58·31-s + 2.69·32-s − 11.9·34-s − 4.89·35-s + 0.840·37-s + 11.9·38-s + 5.63·40-s + 2.51·41-s + ⋯ |
| L(s) = 1 | + 1.76·2-s + 2.12·4-s + 0.447·5-s − 1.85·7-s + 1.99·8-s + 0.790·10-s − 1.54·11-s + 0.0814·13-s − 3.27·14-s + 1.39·16-s − 1.16·17-s + 1.09·19-s + 0.951·20-s − 2.73·22-s − 1.08·23-s + 0.200·25-s + 0.144·26-s − 3.93·28-s − 1.64·29-s − 1.18·31-s + 0.476·32-s − 2.05·34-s − 0.827·35-s + 0.138·37-s + 1.94·38-s + 0.890·40-s + 0.392·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 89 | \( 1 + T \) |
| good | 2 | \( 1 - 2.50T + 2T^{2} \) |
| 7 | \( 1 + 4.89T + 7T^{2} \) |
| 11 | \( 1 + 5.13T + 11T^{2} \) |
| 13 | \( 1 - 0.293T + 13T^{2} \) |
| 17 | \( 1 + 4.78T + 17T^{2} \) |
| 19 | \( 1 - 4.79T + 19T^{2} \) |
| 23 | \( 1 + 5.18T + 23T^{2} \) |
| 29 | \( 1 + 8.87T + 29T^{2} \) |
| 31 | \( 1 + 6.58T + 31T^{2} \) |
| 37 | \( 1 - 0.840T + 37T^{2} \) |
| 41 | \( 1 - 2.51T + 41T^{2} \) |
| 43 | \( 1 - 9.39T + 43T^{2} \) |
| 47 | \( 1 + 8.06T + 47T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 59 | \( 1 + 1.04T + 59T^{2} \) |
| 61 | \( 1 - 6.77T + 61T^{2} \) |
| 67 | \( 1 + 14.0T + 67T^{2} \) |
| 71 | \( 1 + 2.08T + 71T^{2} \) |
| 73 | \( 1 - 6.37T + 73T^{2} \) |
| 79 | \( 1 - 0.566T + 79T^{2} \) |
| 83 | \( 1 + 6.12T + 83T^{2} \) |
| 97 | \( 1 + 3.68T + 97T^{2} \) |
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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
−7.57341485738314105293016898578, −7.14141453540294516122816935253, −6.25724083801928953921005851474, −5.73650817946596154250335579372, −5.26923413147789675021951000129, −4.15179138127968518965138624950, −3.49985222781124143855024423158, −2.72829893867766070638604148700, −2.13208635336916323987514522934, 0,
2.13208635336916323987514522934, 2.72829893867766070638604148700, 3.49985222781124143855024423158, 4.15179138127968518965138624950, 5.26923413147789675021951000129, 5.73650817946596154250335579372, 6.25724083801928953921005851474, 7.14141453540294516122816935253, 7.57341485738314105293016898578
