L(s) = 1 | − 1.73·2-s + 0.999·4-s + 5-s − 1.73·7-s + 1.73·8-s − 1.73·10-s + 3.46·13-s + 2.99·14-s − 5·16-s + 6.92·17-s − 3.46·19-s + 0.999·20-s + 25-s − 5.99·26-s − 1.73·28-s − 8·31-s + 5.19·32-s − 11.9·34-s − 1.73·35-s − 8·37-s + 5.99·38-s + 1.73·40-s − 12.1·41-s + 8.66·43-s − 9·47-s − 4·49-s − 1.73·50-s + ⋯ |
L(s) = 1 | − 1.22·2-s + 0.499·4-s + 0.447·5-s − 0.654·7-s + 0.612·8-s − 0.547·10-s + 0.960·13-s + 0.801·14-s − 1.25·16-s + 1.68·17-s − 0.794·19-s + 0.223·20-s + 0.200·25-s − 1.17·26-s − 0.327·28-s − 1.43·31-s + 0.918·32-s − 2.05·34-s − 0.292·35-s − 1.31·37-s + 0.973·38-s + 0.273·40-s − 1.89·41-s + 1.32·43-s − 1.31·47-s − 0.571·49-s − 0.244·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 1.73T + 2T^{2} \) |
| 7 | \( 1 + 1.73T + 7T^{2} \) |
| 13 | \( 1 - 3.46T + 13T^{2} \) |
| 17 | \( 1 - 6.92T + 17T^{2} \) |
| 19 | \( 1 + 3.46T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 8T + 37T^{2} \) |
| 41 | \( 1 + 12.1T + 41T^{2} \) |
| 43 | \( 1 - 8.66T + 43T^{2} \) |
| 47 | \( 1 + 9T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 8.66T + 61T^{2} \) |
| 67 | \( 1 + 5T + 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 + 3.46T + 83T^{2} \) |
| 89 | \( 1 + 3T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−8.096925519354086749851949809955, −7.14718580513393105353197166477, −6.63549011768834866159462033533, −5.74390795652451704881136372192, −5.08549184928849393062390072085, −3.87505217806754919209178757499, −3.24346262267211109662656867219, −1.96250289561117345818381145766, −1.22535728293711359533461862250, 0,
1.22535728293711359533461862250, 1.96250289561117345818381145766, 3.24346262267211109662656867219, 3.87505217806754919209178757499, 5.08549184928849393062390072085, 5.74390795652451704881136372192, 6.63549011768834866159462033533, 7.14718580513393105353197166477, 8.096925519354086749851949809955