| L(s) = 1 | + 2-s + 3.19·3-s + 4-s + 0.311·5-s + 3.19·6-s + 4.83·7-s + 8-s + 7.18·9-s + 0.311·10-s + 0.258·11-s + 3.19·12-s − 4.86·13-s + 4.83·14-s + 0.993·15-s + 16-s + 1.99·17-s + 7.18·18-s + 2.04·19-s + 0.311·20-s + 15.4·21-s + 0.258·22-s + 23-s + 3.19·24-s − 4.90·25-s − 4.86·26-s + 13.3·27-s + 4.83·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.84·3-s + 0.5·4-s + 0.139·5-s + 1.30·6-s + 1.82·7-s + 0.353·8-s + 2.39·9-s + 0.0984·10-s + 0.0779·11-s + 0.921·12-s − 1.34·13-s + 1.29·14-s + 0.256·15-s + 0.250·16-s + 0.483·17-s + 1.69·18-s + 0.470·19-s + 0.0696·20-s + 3.36·21-s + 0.0551·22-s + 0.208·23-s + 0.651·24-s − 0.980·25-s − 0.953·26-s + 2.56·27-s + 0.913·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.086092724\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.086092724\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 - 3.19T + 3T^{2} \) |
| 5 | \( 1 - 0.311T + 5T^{2} \) |
| 7 | \( 1 - 4.83T + 7T^{2} \) |
| 11 | \( 1 - 0.258T + 11T^{2} \) |
| 13 | \( 1 + 4.86T + 13T^{2} \) |
| 17 | \( 1 - 1.99T + 17T^{2} \) |
| 19 | \( 1 - 2.04T + 19T^{2} \) |
| 29 | \( 1 + 9.68T + 29T^{2} \) |
| 31 | \( 1 + 7.77T + 31T^{2} \) |
| 37 | \( 1 - 9.83T + 37T^{2} \) |
| 41 | \( 1 + 3.56T + 41T^{2} \) |
| 43 | \( 1 + 9.31T + 43T^{2} \) |
| 47 | \( 1 - 4.47T + 47T^{2} \) |
| 53 | \( 1 - 0.968T + 53T^{2} \) |
| 59 | \( 1 - 7.24T + 59T^{2} \) |
| 61 | \( 1 - 9.91T + 61T^{2} \) |
| 67 | \( 1 + 8.65T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 - 1.51T + 73T^{2} \) |
| 79 | \( 1 - 6.76T + 79T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 + 0.559T + 89T^{2} \) |
| 97 | \( 1 + 1.15T + 97T^{2} \) |
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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
−7.963060032573749549251542564134, −7.43539760640334872474364210960, −7.15147824604675841852292633849, −5.61870537463022052143544940756, −5.07957891867756862999453143886, −4.24829716401304520571118180006, −3.71659536277772935275899413431, −2.71743812057537485661962845192, −2.03015451796345946174555756370, −1.50676550173755815421132191369,
1.50676550173755815421132191369, 2.03015451796345946174555756370, 2.71743812057537485661962845192, 3.71659536277772935275899413431, 4.24829716401304520571118180006, 5.07957891867756862999453143886, 5.61870537463022052143544940756, 7.15147824604675841852292633849, 7.43539760640334872474364210960, 7.963060032573749549251542564134
