| L(s) = 1 | − 0.628·2-s − 2.72·3-s − 1.60·4-s + 1.23·5-s + 1.71·6-s − 4.08·7-s + 2.26·8-s + 4.40·9-s − 0.778·10-s − 0.719·11-s + 4.36·12-s + 2.56·14-s − 3.37·15-s + 1.78·16-s − 6.36·17-s − 2.76·18-s + 1.42·19-s − 1.98·20-s + 11.1·21-s + 0.451·22-s + 0.0531·23-s − 6.16·24-s − 3.46·25-s − 3.82·27-s + 6.55·28-s + 3.14·29-s + 2.11·30-s + ⋯ |
| L(s) = 1 | − 0.444·2-s − 1.57·3-s − 0.802·4-s + 0.554·5-s + 0.698·6-s − 1.54·7-s + 0.801·8-s + 1.46·9-s − 0.246·10-s − 0.216·11-s + 1.26·12-s + 0.685·14-s − 0.870·15-s + 0.446·16-s − 1.54·17-s − 0.652·18-s + 0.327·19-s − 0.444·20-s + 2.42·21-s + 0.0963·22-s + 0.0110·23-s − 1.25·24-s − 0.692·25-s − 0.736·27-s + 1.23·28-s + 0.584·29-s + 0.386·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1189323576\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1189323576\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 0.628T + 2T^{2} \) |
| 3 | \( 1 + 2.72T + 3T^{2} \) |
| 5 | \( 1 - 1.23T + 5T^{2} \) |
| 7 | \( 1 + 4.08T + 7T^{2} \) |
| 11 | \( 1 + 0.719T + 11T^{2} \) |
| 17 | \( 1 + 6.36T + 17T^{2} \) |
| 19 | \( 1 - 1.42T + 19T^{2} \) |
| 23 | \( 1 - 0.0531T + 23T^{2} \) |
| 29 | \( 1 - 3.14T + 29T^{2} \) |
| 37 | \( 1 + 0.646T + 37T^{2} \) |
| 41 | \( 1 + 3.47T + 41T^{2} \) |
| 43 | \( 1 + 8.98T + 43T^{2} \) |
| 47 | \( 1 - 2.63T + 47T^{2} \) |
| 53 | \( 1 - 7.67T + 53T^{2} \) |
| 59 | \( 1 - 7.53T + 59T^{2} \) |
| 61 | \( 1 - 7.43T + 61T^{2} \) |
| 67 | \( 1 + 12.2T + 67T^{2} \) |
| 71 | \( 1 + 15.7T + 71T^{2} \) |
| 73 | \( 1 + 12.8T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 + 15.0T + 83T^{2} \) |
| 89 | \( 1 + 5.31T + 89T^{2} \) |
| 97 | \( 1 - 4.59T + 97T^{2} \) |
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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
−8.402010469258533864839475168629, −7.09229383023528776251296715456, −6.81224171824915171456002364734, −5.89455295541986665567245816752, −5.55086967515005016382986677895, −4.61654420122044578737462617490, −3.98226280419409669923558508185, −2.80894434791817505878642191434, −1.46957254388996913973469518690, −0.22560603015122781143968037998,
0.22560603015122781143968037998, 1.46957254388996913973469518690, 2.80894434791817505878642191434, 3.98226280419409669923558508185, 4.61654420122044578737462617490, 5.55086967515005016382986677895, 5.89455295541986665567245816752, 6.81224171824915171456002364734, 7.09229383023528776251296715456, 8.402010469258533864839475168629
