| L(s) = 1 | − 2-s + 2.17·3-s + 4-s − 0.950·5-s − 2.17·6-s − 2.22·7-s − 8-s + 1.74·9-s + 0.950·10-s − 11-s + 2.17·12-s − 6.34·13-s + 2.22·14-s − 2.07·15-s + 16-s − 1.98·17-s − 1.74·18-s + 1.57·19-s − 0.950·20-s − 4.85·21-s + 22-s + 2.04·23-s − 2.17·24-s − 4.09·25-s + 6.34·26-s − 2.73·27-s − 2.22·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.25·3-s + 0.5·4-s − 0.425·5-s − 0.889·6-s − 0.842·7-s − 0.353·8-s + 0.582·9-s + 0.300·10-s − 0.301·11-s + 0.628·12-s − 1.76·13-s + 0.595·14-s − 0.534·15-s + 0.250·16-s − 0.481·17-s − 0.411·18-s + 0.361·19-s − 0.212·20-s − 1.05·21-s + 0.213·22-s + 0.426·23-s − 0.444·24-s − 0.819·25-s + 1.24·26-s − 0.525·27-s − 0.421·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 946 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 946 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 3 | \( 1 - 2.17T + 3T^{2} \) |
| 5 | \( 1 + 0.950T + 5T^{2} \) |
| 7 | \( 1 + 2.22T + 7T^{2} \) |
| 13 | \( 1 + 6.34T + 13T^{2} \) |
| 17 | \( 1 + 1.98T + 17T^{2} \) |
| 19 | \( 1 - 1.57T + 19T^{2} \) |
| 23 | \( 1 - 2.04T + 23T^{2} \) |
| 29 | \( 1 + 8.06T + 29T^{2} \) |
| 31 | \( 1 - 2.86T + 31T^{2} \) |
| 37 | \( 1 - 3.33T + 37T^{2} \) |
| 41 | \( 1 + 1.92T + 41T^{2} \) |
| 47 | \( 1 + 0.653T + 47T^{2} \) |
| 53 | \( 1 - 3.88T + 53T^{2} \) |
| 59 | \( 1 + 5.84T + 59T^{2} \) |
| 61 | \( 1 - 0.832T + 61T^{2} \) |
| 67 | \( 1 - 14.2T + 67T^{2} \) |
| 71 | \( 1 + 16.8T + 71T^{2} \) |
| 73 | \( 1 - 14.7T + 73T^{2} \) |
| 79 | \( 1 - 8.54T + 79T^{2} \) |
| 83 | \( 1 - 3.10T + 83T^{2} \) |
| 89 | \( 1 - 5.12T + 89T^{2} \) |
| 97 | \( 1 + 3.34T + 97T^{2} \) |
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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
−9.559619045492341802354068750273, −8.920953928404605687092294924728, −7.86019755333618217693261171224, −7.51436579704281526098565953150, −6.55703297007710717806139770800, −5.23008740861388867736960841190, −3.88037738862858524441535579229, −2.92220884073749853536853339060, −2.15496333735879494950448345519, 0,
2.15496333735879494950448345519, 2.92220884073749853536853339060, 3.88037738862858524441535579229, 5.23008740861388867736960841190, 6.55703297007710717806139770800, 7.51436579704281526098565953150, 7.86019755333618217693261171224, 8.920953928404605687092294924728, 9.559619045492341802354068750273
