| L(s) = 1 | − 0.339·2-s − 1.88·4-s + 2.32·5-s + 3.72·7-s + 1.31·8-s − 0.789·10-s − 11-s − 5.91·13-s − 1.26·14-s + 3.32·16-s + 6.90·17-s + 3.48·19-s − 4.39·20-s + 0.339·22-s + 0.198·23-s + 0.425·25-s + 2.00·26-s − 7.01·28-s − 9.69·29-s − 6.24·31-s − 3.76·32-s − 2.33·34-s + 8.66·35-s − 9.17·37-s − 1.18·38-s + 3.06·40-s − 7.54·41-s + ⋯ |
| L(s) = 1 | − 0.239·2-s − 0.942·4-s + 1.04·5-s + 1.40·7-s + 0.465·8-s − 0.249·10-s − 0.301·11-s − 1.63·13-s − 0.337·14-s + 0.830·16-s + 1.67·17-s + 0.799·19-s − 0.981·20-s + 0.0722·22-s + 0.0412·23-s + 0.0850·25-s + 0.393·26-s − 1.32·28-s − 1.80·29-s − 1.12·31-s − 0.664·32-s − 0.401·34-s + 1.46·35-s − 1.50·37-s − 0.191·38-s + 0.485·40-s − 1.17·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8019 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8019 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 0.339T + 2T^{2} \) |
| 5 | \( 1 - 2.32T + 5T^{2} \) |
| 7 | \( 1 - 3.72T + 7T^{2} \) |
| 13 | \( 1 + 5.91T + 13T^{2} \) |
| 17 | \( 1 - 6.90T + 17T^{2} \) |
| 19 | \( 1 - 3.48T + 19T^{2} \) |
| 23 | \( 1 - 0.198T + 23T^{2} \) |
| 29 | \( 1 + 9.69T + 29T^{2} \) |
| 31 | \( 1 + 6.24T + 31T^{2} \) |
| 37 | \( 1 + 9.17T + 37T^{2} \) |
| 41 | \( 1 + 7.54T + 41T^{2} \) |
| 43 | \( 1 + 6.12T + 43T^{2} \) |
| 47 | \( 1 - 3.60T + 47T^{2} \) |
| 53 | \( 1 + 1.27T + 53T^{2} \) |
| 59 | \( 1 - 0.763T + 59T^{2} \) |
| 61 | \( 1 + 1.70T + 61T^{2} \) |
| 67 | \( 1 - 0.138T + 67T^{2} \) |
| 71 | \( 1 + 5.67T + 71T^{2} \) |
| 73 | \( 1 - 11.8T + 73T^{2} \) |
| 79 | \( 1 + 9.60T + 79T^{2} \) |
| 83 | \( 1 + 13.1T + 83T^{2} \) |
| 89 | \( 1 + 5.90T + 89T^{2} \) |
| 97 | \( 1 + 12.5T + 97T^{2} \) |
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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
−7.52884832295621143413443918524, −7.15241518082572113398759529135, −5.61482159595427201949423314600, −5.33207908867195310800914205724, −5.03609206402669500758086585548, −3.98016630680745531461203102746, −3.08917167478857538446586280097, −1.88153482457042826626826278175, −1.45832604742026690792110209442, 0,
1.45832604742026690792110209442, 1.88153482457042826626826278175, 3.08917167478857538446586280097, 3.98016630680745531461203102746, 5.03609206402669500758086585548, 5.33207908867195310800914205724, 5.61482159595427201949423314600, 7.15241518082572113398759529135, 7.52884832295621143413443918524
