| L(s) = 1 | − 2.27·2-s + 0.713·3-s + 3.19·4-s + 3.36·5-s − 1.62·6-s − 2.72·8-s − 2.49·9-s − 7.66·10-s − 2.45·11-s + 2.27·12-s − 1.53·13-s + 2.40·15-s − 0.186·16-s + 2.89·17-s + 5.67·18-s − 1.01·19-s + 10.7·20-s + 5.59·22-s − 4.93·23-s − 1.94·24-s + 6.32·25-s + 3.48·26-s − 3.91·27-s − 8.76·29-s − 5.47·30-s + 5.91·31-s + 5.86·32-s + ⋯ |
| L(s) = 1 | − 1.61·2-s + 0.412·3-s + 1.59·4-s + 1.50·5-s − 0.663·6-s − 0.962·8-s − 0.830·9-s − 2.42·10-s − 0.740·11-s + 0.658·12-s − 0.424·13-s + 0.620·15-s − 0.0465·16-s + 0.702·17-s + 1.33·18-s − 0.233·19-s + 2.40·20-s + 1.19·22-s − 1.02·23-s − 0.396·24-s + 1.26·25-s + 0.684·26-s − 0.754·27-s − 1.62·29-s − 0.999·30-s + 1.06·31-s + 1.03·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 + 2.27T + 2T^{2} \) |
| 3 | \( 1 - 0.713T + 3T^{2} \) |
| 5 | \( 1 - 3.36T + 5T^{2} \) |
| 11 | \( 1 + 2.45T + 11T^{2} \) |
| 13 | \( 1 + 1.53T + 13T^{2} \) |
| 17 | \( 1 - 2.89T + 17T^{2} \) |
| 19 | \( 1 + 1.01T + 19T^{2} \) |
| 23 | \( 1 + 4.93T + 23T^{2} \) |
| 29 | \( 1 + 8.76T + 29T^{2} \) |
| 31 | \( 1 - 5.91T + 31T^{2} \) |
| 37 | \( 1 + 8.45T + 37T^{2} \) |
| 43 | \( 1 + 5.59T + 43T^{2} \) |
| 47 | \( 1 + 5.14T + 47T^{2} \) |
| 53 | \( 1 + 7.81T + 53T^{2} \) |
| 59 | \( 1 - 6.63T + 59T^{2} \) |
| 61 | \( 1 + 9.20T + 61T^{2} \) |
| 67 | \( 1 + 9.54T + 67T^{2} \) |
| 71 | \( 1 + 7.18T + 71T^{2} \) |
| 73 | \( 1 - 2.87T + 73T^{2} \) |
| 79 | \( 1 - 3.80T + 79T^{2} \) |
| 83 | \( 1 - 14.4T + 83T^{2} \) |
| 89 | \( 1 - 7.01T + 89T^{2} \) |
| 97 | \( 1 - 5.72T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.919592248123247968464764528601, −8.117743453834666632974697387914, −7.59361789435721870486929657385, −6.51586966729166003622696741296, −5.81803293895227942723089383635, −4.99689670574850358915401901200, −3.23988246906646490596249592983, −2.27319447070732448156102720623, −1.67764295671640599868099466285, 0,
1.67764295671640599868099466285, 2.27319447070732448156102720623, 3.23988246906646490596249592983, 4.99689670574850358915401901200, 5.81803293895227942723089383635, 6.51586966729166003622696741296, 7.59361789435721870486929657385, 8.117743453834666632974697387914, 8.919592248123247968464764528601
