| L(s) = 1 | + 1.76·2-s + 1.12·4-s + 5-s − 4.90·7-s − 1.54·8-s + 1.76·10-s + 1.01·11-s + 5.60·13-s − 8.67·14-s − 4.98·16-s + 1.50·17-s − 0.730·19-s + 1.12·20-s + 1.80·22-s − 3.99·23-s + 25-s + 9.90·26-s − 5.53·28-s + 5.27·29-s − 0.424·31-s − 5.72·32-s + 2.65·34-s − 4.90·35-s − 8.45·37-s − 1.29·38-s − 1.54·40-s − 3.05·41-s + ⋯ |
| L(s) = 1 | + 1.25·2-s + 0.563·4-s + 0.447·5-s − 1.85·7-s − 0.545·8-s + 0.559·10-s + 0.307·11-s + 1.55·13-s − 2.31·14-s − 1.24·16-s + 0.364·17-s − 0.167·19-s + 0.252·20-s + 0.384·22-s − 0.833·23-s + 0.200·25-s + 1.94·26-s − 1.04·28-s + 0.980·29-s − 0.0762·31-s − 1.01·32-s + 0.455·34-s − 0.829·35-s − 1.38·37-s − 0.209·38-s − 0.243·40-s − 0.476·41-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 | \( 1 - 1.76T + 2T^{2} \) |
| 7 | \( 1 + 4.90T + 7T^{2} \) |
| 11 | \( 1 - 1.01T + 11T^{2} \) |
| 13 | \( 1 - 5.60T + 13T^{2} \) |
| 17 | \( 1 - 1.50T + 17T^{2} \) |
| 19 | \( 1 + 0.730T + 19T^{2} \) |
| 23 | \( 1 + 3.99T + 23T^{2} \) |
| 29 | \( 1 - 5.27T + 29T^{2} \) |
| 31 | \( 1 + 0.424T + 31T^{2} \) |
| 37 | \( 1 + 8.45T + 37T^{2} \) |
| 41 | \( 1 + 3.05T + 41T^{2} \) |
| 43 | \( 1 + 11.6T + 43T^{2} \) |
| 47 | \( 1 + 9.79T + 47T^{2} \) |
| 53 | \( 1 + 11.5T + 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 - 0.584T + 61T^{2} \) |
| 67 | \( 1 + 13.3T + 67T^{2} \) |
| 71 | \( 1 + 8.47T + 71T^{2} \) |
| 73 | \( 1 - 16.4T + 73T^{2} \) |
| 79 | \( 1 - 2.07T + 79T^{2} \) |
| 83 | \( 1 + 2.20T + 83T^{2} \) |
| 97 | \( 1 - 2.83T + 97T^{2} \) |
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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
−8.152367010294427572536802138908, −6.69751318294210292367927634922, −6.47958187320553943962664630568, −5.94886812178574103073411969920, −5.12476081760429346608693585459, −4.13187848289469778643257981523, −3.31675914623226958428103316571, −3.11240819113432426041603807907, −1.65689410790819122792879919734, 0,
1.65689410790819122792879919734, 3.11240819113432426041603807907, 3.31675914623226958428103316571, 4.13187848289469778643257981523, 5.12476081760429346608693585459, 5.94886812178574103073411969920, 6.47958187320553943962664630568, 6.69751318294210292367927634922, 8.152367010294427572536802138908
