| L(s) = 1 | + 2-s − 2.62·3-s + 4-s − 2.01·5-s − 2.62·6-s + 0.869·7-s + 8-s + 3.89·9-s − 2.01·10-s − 3.06·11-s − 2.62·12-s + 4.63·13-s + 0.869·14-s + 5.30·15-s + 16-s − 0.775·17-s + 3.89·18-s − 4.44·19-s − 2.01·20-s − 2.28·21-s − 3.06·22-s − 0.835·23-s − 2.62·24-s − 0.927·25-s + 4.63·26-s − 2.36·27-s + 0.869·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.51·3-s + 0.5·4-s − 0.902·5-s − 1.07·6-s + 0.328·7-s + 0.353·8-s + 1.29·9-s − 0.638·10-s − 0.924·11-s − 0.758·12-s + 1.28·13-s + 0.232·14-s + 1.36·15-s + 0.250·16-s − 0.187·17-s + 0.919·18-s − 1.01·19-s − 0.451·20-s − 0.498·21-s − 0.653·22-s − 0.174·23-s − 0.536·24-s − 0.185·25-s + 0.908·26-s − 0.454·27-s + 0.164·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 2011 | \( 1 - T \) |
| good | 3 | \( 1 + 2.62T + 3T^{2} \) |
| 5 | \( 1 + 2.01T + 5T^{2} \) |
| 7 | \( 1 - 0.869T + 7T^{2} \) |
| 11 | \( 1 + 3.06T + 11T^{2} \) |
| 13 | \( 1 - 4.63T + 13T^{2} \) |
| 17 | \( 1 + 0.775T + 17T^{2} \) |
| 19 | \( 1 + 4.44T + 19T^{2} \) |
| 23 | \( 1 + 0.835T + 23T^{2} \) |
| 29 | \( 1 - 7.31T + 29T^{2} \) |
| 31 | \( 1 + 1.45T + 31T^{2} \) |
| 37 | \( 1 - 9.35T + 37T^{2} \) |
| 41 | \( 1 - 1.66T + 41T^{2} \) |
| 43 | \( 1 + 9.08T + 43T^{2} \) |
| 47 | \( 1 + 1.03T + 47T^{2} \) |
| 53 | \( 1 - 4.38T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 + 0.898T + 61T^{2} \) |
| 67 | \( 1 - 4.69T + 67T^{2} \) |
| 71 | \( 1 - 5.82T + 71T^{2} \) |
| 73 | \( 1 - 0.985T + 73T^{2} \) |
| 79 | \( 1 + 8.32T + 79T^{2} \) |
| 83 | \( 1 + 6.68T + 83T^{2} \) |
| 89 | \( 1 + 5.26T + 89T^{2} \) |
| 97 | \( 1 + 6.27T + 97T^{2} \) |
| 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
−8.160964864041866805456219851929, −7.03496199651351415277095474250, −6.43799580720961952677959816797, −5.80626563612913254803188561602, −5.07857089336844384236290631955, −4.39170583711417478771468557735, −3.78180293201483464990698484311, −2.57761190210306903831537957155, −1.21487920471421442906729718034, 0,
1.21487920471421442906729718034, 2.57761190210306903831537957155, 3.78180293201483464990698484311, 4.39170583711417478771468557735, 5.07857089336844384236290631955, 5.80626563612913254803188561602, 6.43799580720961952677959816797, 7.03496199651351415277095474250, 8.160964864041866805456219851929
