L(s) = 1 | − 1.32·2-s − 0.239·4-s − 2.32·5-s + 7-s + 2.97·8-s + 3.08·10-s + 4.53·13-s − 1.32·14-s − 3.46·16-s + 3.29·17-s + 1.08·19-s + 0.556·20-s − 6.29·23-s + 0.414·25-s − 6.02·26-s − 0.239·28-s − 3.16·29-s − 2.97·31-s − 1.34·32-s − 4.37·34-s − 2.32·35-s + 1.70·37-s − 1.44·38-s − 6.91·40-s − 4.90·41-s + 8.38·43-s + 8.35·46-s + ⋯ |
L(s) = 1 | − 0.938·2-s − 0.119·4-s − 1.04·5-s + 0.377·7-s + 1.05·8-s + 0.976·10-s + 1.25·13-s − 0.354·14-s − 0.866·16-s + 0.799·17-s + 0.249·19-s + 0.124·20-s − 1.31·23-s + 0.0829·25-s − 1.18·26-s − 0.0452·28-s − 0.587·29-s − 0.533·31-s − 0.237·32-s − 0.750·34-s − 0.393·35-s + 0.280·37-s − 0.234·38-s − 1.09·40-s − 0.766·41-s + 1.27·43-s + 1.23·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 1.32T + 2T^{2} \) |
| 5 | \( 1 + 2.32T + 5T^{2} \) |
| 13 | \( 1 - 4.53T + 13T^{2} \) |
| 17 | \( 1 - 3.29T + 17T^{2} \) |
| 19 | \( 1 - 1.08T + 19T^{2} \) |
| 23 | \( 1 + 6.29T + 23T^{2} \) |
| 29 | \( 1 + 3.16T + 29T^{2} \) |
| 31 | \( 1 + 2.97T + 31T^{2} \) |
| 37 | \( 1 - 1.70T + 37T^{2} \) |
| 41 | \( 1 + 4.90T + 41T^{2} \) |
| 43 | \( 1 - 8.38T + 43T^{2} \) |
| 47 | \( 1 - 4.69T + 47T^{2} \) |
| 53 | \( 1 - 4.98T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 - 1.74T + 61T^{2} \) |
| 67 | \( 1 + 9.44T + 67T^{2} \) |
| 71 | \( 1 + 16.1T + 71T^{2} \) |
| 73 | \( 1 + 14.4T + 73T^{2} \) |
| 79 | \( 1 - 12.8T + 79T^{2} \) |
| 83 | \( 1 - 0.205T + 83T^{2} \) |
| 89 | \( 1 + 3.93T + 89T^{2} \) |
| 97 | \( 1 - 13.0T + 97T^{2} \) |
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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
−7.53971807111090548841845973820, −7.42142705159298384363279553925, −6.12593948928677484687189863009, −5.53960993290554746717526048369, −4.44462856542187592648290897767, −3.99578983023740806329280717855, −3.26448269797650438222672774633, −1.88012915503410521712777542921, −1.05795096852063193380048276502, 0,
1.05795096852063193380048276502, 1.88012915503410521712777542921, 3.26448269797650438222672774633, 3.99578983023740806329280717855, 4.44462856542187592648290897767, 5.53960993290554746717526048369, 6.12593948928677484687189863009, 7.42142705159298384363279553925, 7.53971807111090548841845973820