| L(s) = 1 | + 2-s + 1.13·3-s + 4-s + 5-s + 1.13·6-s − 0.772·7-s + 8-s − 1.72·9-s + 10-s − 5.33·11-s + 1.13·12-s − 2.27·13-s − 0.772·14-s + 1.13·15-s + 16-s − 1.51·17-s − 1.72·18-s − 4.75·19-s + 20-s − 0.873·21-s − 5.33·22-s + 3.57·23-s + 1.13·24-s + 25-s − 2.27·26-s − 5.33·27-s − 0.772·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.652·3-s + 0.5·4-s + 0.447·5-s + 0.461·6-s − 0.292·7-s + 0.353·8-s − 0.574·9-s + 0.316·10-s − 1.60·11-s + 0.326·12-s − 0.630·13-s − 0.206·14-s + 0.291·15-s + 0.250·16-s − 0.367·17-s − 0.405·18-s − 1.09·19-s + 0.223·20-s − 0.190·21-s − 1.13·22-s + 0.745·23-s + 0.230·24-s + 0.200·25-s − 0.445·26-s − 1.02·27-s − 0.146·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
| good | 3 | \( 1 - 1.13T + 3T^{2} \) |
| 7 | \( 1 + 0.772T + 7T^{2} \) |
| 11 | \( 1 + 5.33T + 11T^{2} \) |
| 13 | \( 1 + 2.27T + 13T^{2} \) |
| 17 | \( 1 + 1.51T + 17T^{2} \) |
| 19 | \( 1 + 4.75T + 19T^{2} \) |
| 23 | \( 1 - 3.57T + 23T^{2} \) |
| 29 | \( 1 - 0.625T + 29T^{2} \) |
| 31 | \( 1 - 4.86T + 31T^{2} \) |
| 37 | \( 1 + 4.40T + 37T^{2} \) |
| 41 | \( 1 - 0.597T + 41T^{2} \) |
| 43 | \( 1 + 3.51T + 43T^{2} \) |
| 47 | \( 1 - 1.44T + 47T^{2} \) |
| 53 | \( 1 + 1.59T + 53T^{2} \) |
| 59 | \( 1 + 9.14T + 59T^{2} \) |
| 61 | \( 1 - 3.55T + 61T^{2} \) |
| 67 | \( 1 - 9.65T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 + 0.0209T + 79T^{2} \) |
| 83 | \( 1 + 15.8T + 83T^{2} \) |
| 89 | \( 1 - 10.7T + 89T^{2} \) |
| 97 | \( 1 + 14.7T + 97T^{2} \) |
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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
−8.135403522065140978164726252464, −7.31370796331475037683526728851, −6.54402585449299856329768398630, −5.74062295201829733972171457498, −5.07128128860264525033869652829, −4.35336834767940974892376204003, −3.09911781141934324790826952024, −2.72856373318853933302494166791, −1.91628649896114209265902253566, 0,
1.91628649896114209265902253566, 2.72856373318853933302494166791, 3.09911781141934324790826952024, 4.35336834767940974892376204003, 5.07128128860264525033869652829, 5.74062295201829733972171457498, 6.54402585449299856329768398630, 7.31370796331475037683526728851, 8.135403522065140978164726252464
