| L(s) = 1 | − 2.24·3-s + 1.47·5-s + 2.05·9-s − 0.494·11-s − 5.52·13-s − 3.30·15-s + 4.25·17-s + 0.960·19-s − 2.30·23-s − 2.83·25-s + 2.11·27-s + 1.25·29-s + 8.26·31-s + 1.11·33-s − 8.22·37-s + 12.4·39-s − 41-s + 5.33·43-s + 3.02·45-s − 3.66·47-s − 9.57·51-s + 8.54·53-s − 0.727·55-s − 2.16·57-s − 1.47·59-s + 6.23·61-s − 8.12·65-s + ⋯ |
| L(s) = 1 | − 1.29·3-s + 0.657·5-s + 0.686·9-s − 0.149·11-s − 1.53·13-s − 0.854·15-s + 1.03·17-s + 0.220·19-s − 0.479·23-s − 0.567·25-s + 0.407·27-s + 0.233·29-s + 1.48·31-s + 0.193·33-s − 1.35·37-s + 1.98·39-s − 0.156·41-s + 0.813·43-s + 0.451·45-s − 0.534·47-s − 1.34·51-s + 1.17·53-s − 0.0980·55-s − 0.286·57-s − 0.191·59-s + 0.798·61-s − 1.00·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 + 2.24T + 3T^{2} \) |
| 5 | \( 1 - 1.47T + 5T^{2} \) |
| 11 | \( 1 + 0.494T + 11T^{2} \) |
| 13 | \( 1 + 5.52T + 13T^{2} \) |
| 17 | \( 1 - 4.25T + 17T^{2} \) |
| 19 | \( 1 - 0.960T + 19T^{2} \) |
| 23 | \( 1 + 2.30T + 23T^{2} \) |
| 29 | \( 1 - 1.25T + 29T^{2} \) |
| 31 | \( 1 - 8.26T + 31T^{2} \) |
| 37 | \( 1 + 8.22T + 37T^{2} \) |
| 43 | \( 1 - 5.33T + 43T^{2} \) |
| 47 | \( 1 + 3.66T + 47T^{2} \) |
| 53 | \( 1 - 8.54T + 53T^{2} \) |
| 59 | \( 1 + 1.47T + 59T^{2} \) |
| 61 | \( 1 - 6.23T + 61T^{2} \) |
| 67 | \( 1 - 3.91T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 + 0.775T + 73T^{2} \) |
| 79 | \( 1 + 9.98T + 79T^{2} \) |
| 83 | \( 1 - 7.68T + 83T^{2} \) |
| 89 | \( 1 - 6.83T + 89T^{2} \) |
| 97 | \( 1 - 6.83T + 97T^{2} \) |
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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
−7.36571415265271803823875969508, −6.64809589141939398363944825714, −6.01371827870778193354959061047, −5.35592654559401083223954882413, −5.01008440177361834129606318558, −4.12530761744064872427324679675, −2.98315702628255083646813287090, −2.16844347321112720247053370903, −1.07199718996175142667775293852, 0,
1.07199718996175142667775293852, 2.16844347321112720247053370903, 2.98315702628255083646813287090, 4.12530761744064872427324679675, 5.01008440177361834129606318558, 5.35592654559401083223954882413, 6.01371827870778193354959061047, 6.64809589141939398363944825714, 7.36571415265271803823875969508
