L(s) = 1 | − 2.17·2-s + 2.70·4-s + 0.539·5-s + 0.630·7-s − 1.53·8-s − 1.17·10-s − 11-s + 13-s − 1.36·14-s − 2.07·16-s − 1.90·17-s − 4.04·19-s + 1.46·20-s + 2.17·22-s − 1.36·23-s − 4.70·25-s − 2.17·26-s + 1.70·28-s + 2.24·29-s − 1.46·31-s + 7.58·32-s + 4.14·34-s + 0.340·35-s − 5.07·37-s + 8.78·38-s − 0.829·40-s + 2.04·41-s + ⋯ |
L(s) = 1 | − 1.53·2-s + 1.35·4-s + 0.241·5-s + 0.238·7-s − 0.544·8-s − 0.370·10-s − 0.301·11-s + 0.277·13-s − 0.365·14-s − 0.519·16-s − 0.462·17-s − 0.929·19-s + 0.326·20-s + 0.462·22-s − 0.285·23-s − 0.941·25-s − 0.425·26-s + 0.323·28-s + 0.417·29-s − 0.262·31-s + 1.34·32-s + 0.710·34-s + 0.0574·35-s − 0.834·37-s + 1.42·38-s − 0.131·40-s + 0.320·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 + 2.17T + 2T^{2} \) |
| 5 | \( 1 - 0.539T + 5T^{2} \) |
| 7 | \( 1 - 0.630T + 7T^{2} \) |
| 17 | \( 1 + 1.90T + 17T^{2} \) |
| 19 | \( 1 + 4.04T + 19T^{2} \) |
| 23 | \( 1 + 1.36T + 23T^{2} \) |
| 29 | \( 1 - 2.24T + 29T^{2} \) |
| 31 | \( 1 + 1.46T + 31T^{2} \) |
| 37 | \( 1 + 5.07T + 37T^{2} \) |
| 41 | \( 1 - 2.04T + 41T^{2} \) |
| 43 | \( 1 - 0.986T + 43T^{2} \) |
| 47 | \( 1 + 3.26T + 47T^{2} \) |
| 53 | \( 1 + 9.91T + 53T^{2} \) |
| 59 | \( 1 - 3.60T + 59T^{2} \) |
| 61 | \( 1 - 3.41T + 61T^{2} \) |
| 67 | \( 1 - 5.95T + 67T^{2} \) |
| 71 | \( 1 - 7.75T + 71T^{2} \) |
| 73 | \( 1 - 9.46T + 73T^{2} \) |
| 79 | \( 1 - 3.90T + 79T^{2} \) |
| 83 | \( 1 + 2.34T + 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 + 5.07T + 97T^{2} \) |
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
−9.344486466970468732663024091127, −8.323455924878957397445492601394, −8.067794100734637336389997041553, −6.95529805458360708845213016987, −6.29263292776516547968125701464, −5.11027508040897638947445301622, −3.97191252601140782312681217033, −2.45978922854035998292899685183, −1.55326166097877672247257864520, 0,
1.55326166097877672247257864520, 2.45978922854035998292899685183, 3.97191252601140782312681217033, 5.11027508040897638947445301622, 6.29263292776516547968125701464, 6.95529805458360708845213016987, 8.067794100734637336389997041553, 8.323455924878957397445492601394, 9.344486466970468732663024091127