| L(s) = 1 | + 0.558·2-s + 3-s − 1.68·4-s − 0.297·5-s + 0.558·6-s − 2.05·8-s + 9-s − 0.165·10-s + 1.45·11-s − 1.68·12-s − 5.31·13-s − 0.297·15-s + 2.22·16-s − 6.78·17-s + 0.558·18-s + 4.21·19-s + 0.501·20-s + 0.815·22-s + 9.21·23-s − 2.05·24-s − 4.91·25-s − 2.96·26-s + 27-s + 7.74·29-s − 0.165·30-s + 7.15·31-s + 5.36·32-s + ⋯ |
| L(s) = 1 | + 0.394·2-s + 0.577·3-s − 0.844·4-s − 0.132·5-s + 0.227·6-s − 0.728·8-s + 0.333·9-s − 0.0524·10-s + 0.440·11-s − 0.487·12-s − 1.47·13-s − 0.0767·15-s + 0.556·16-s − 1.64·17-s + 0.131·18-s + 0.966·19-s + 0.112·20-s + 0.173·22-s + 1.92·23-s − 0.420·24-s − 0.982·25-s − 0.582·26-s + 0.192·27-s + 1.43·29-s − 0.0302·30-s + 1.28·31-s + 0.947·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 - 0.558T + 2T^{2} \) |
| 5 | \( 1 + 0.297T + 5T^{2} \) |
| 11 | \( 1 - 1.45T + 11T^{2} \) |
| 13 | \( 1 + 5.31T + 13T^{2} \) |
| 17 | \( 1 + 6.78T + 17T^{2} \) |
| 19 | \( 1 - 4.21T + 19T^{2} \) |
| 23 | \( 1 - 9.21T + 23T^{2} \) |
| 29 | \( 1 - 7.74T + 29T^{2} \) |
| 31 | \( 1 - 7.15T + 31T^{2} \) |
| 37 | \( 1 - 1.91T + 37T^{2} \) |
| 43 | \( 1 + 12.3T + 43T^{2} \) |
| 47 | \( 1 - 0.223T + 47T^{2} \) |
| 53 | \( 1 + 3.99T + 53T^{2} \) |
| 59 | \( 1 + 5.57T + 59T^{2} \) |
| 61 | \( 1 + 5.08T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 + 9.53T + 71T^{2} \) |
| 73 | \( 1 - 7.16T + 73T^{2} \) |
| 79 | \( 1 - 2.54T + 79T^{2} \) |
| 83 | \( 1 + 5.99T + 83T^{2} \) |
| 89 | \( 1 + 15.5T + 89T^{2} \) |
| 97 | \( 1 + 10.7T + 97T^{2} \) |
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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
−7.80552412963303960476780664105, −6.97139174270260749794601616466, −6.42489069161421788680937355456, −5.28031541424893575107388928481, −4.65022692063047127757691915492, −4.29065004277547748571190532935, −3.09245435408103861654310265663, −2.70811953881077984111478698678, −1.32470957972900123510121343977, 0,
1.32470957972900123510121343977, 2.70811953881077984111478698678, 3.09245435408103861654310265663, 4.29065004277547748571190532935, 4.65022692063047127757691915492, 5.28031541424893575107388928481, 6.42489069161421788680937355456, 6.97139174270260749794601616466, 7.80552412963303960476780664105
