| L(s) = 1 | + 2.93·5-s + 7-s − 5.73·11-s + 2.54·13-s + 2.34·17-s + 19-s + 2.32·23-s + 3.60·25-s − 6.47·29-s − 9.87·31-s + 2.93·35-s − 3.68·37-s − 10.6·41-s − 4.08·43-s − 4.29·47-s + 49-s − 6.47·53-s − 16.8·55-s − 7.07·59-s − 2·61-s + 7.47·65-s + 2.95·67-s − 8.98·71-s + 7.86·73-s − 5.73·77-s − 2.32·79-s + 3.29·83-s + ⋯ |
| L(s) = 1 | + 1.31·5-s + 0.377·7-s − 1.73·11-s + 0.707·13-s + 0.568·17-s + 0.229·19-s + 0.485·23-s + 0.720·25-s − 1.20·29-s − 1.77·31-s + 0.495·35-s − 0.606·37-s − 1.66·41-s − 0.623·43-s − 0.626·47-s + 0.142·49-s − 0.888·53-s − 2.26·55-s − 0.921·59-s − 0.256·61-s + 0.927·65-s + 0.360·67-s − 1.06·71-s + 0.920·73-s − 0.653·77-s − 0.261·79-s + 0.361·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 2.93T + 5T^{2} \) |
| 11 | \( 1 + 5.73T + 11T^{2} \) |
| 13 | \( 1 - 2.54T + 13T^{2} \) |
| 17 | \( 1 - 2.34T + 17T^{2} \) |
| 23 | \( 1 - 2.32T + 23T^{2} \) |
| 29 | \( 1 + 6.47T + 29T^{2} \) |
| 31 | \( 1 + 9.87T + 31T^{2} \) |
| 37 | \( 1 + 3.68T + 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 + 4.08T + 43T^{2} \) |
| 47 | \( 1 + 4.29T + 47T^{2} \) |
| 53 | \( 1 + 6.47T + 53T^{2} \) |
| 59 | \( 1 + 7.07T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 2.95T + 67T^{2} \) |
| 71 | \( 1 + 8.98T + 71T^{2} \) |
| 73 | \( 1 - 7.86T + 73T^{2} \) |
| 79 | \( 1 + 2.32T + 79T^{2} \) |
| 83 | \( 1 - 3.29T + 83T^{2} \) |
| 89 | \( 1 - 11.7T + 89T^{2} \) |
| 97 | \( 1 + 1.46T + 97T^{2} \) |
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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
−7.44322675420261084642763948574, −6.59349191968873153199911069084, −5.81411799046901910877836848022, −5.28039826057890591302003211539, −4.97473299909012772709611682502, −3.63451380506711449412181397959, −3.00361432491260590772426569744, −1.99721350111461489214513354938, −1.52853704186141024459607808906, 0,
1.52853704186141024459607808906, 1.99721350111461489214513354938, 3.00361432491260590772426569744, 3.63451380506711449412181397959, 4.97473299909012772709611682502, 5.28039826057890591302003211539, 5.81411799046901910877836848022, 6.59349191968873153199911069084, 7.44322675420261084642763948574
