| L(s) = 1 | + 2-s + 4-s − 2.95·5-s + 0.794·7-s + 8-s − 2.95·10-s + 3.15·11-s − 0.00112·13-s + 0.794·14-s + 16-s − 4.32·17-s + 3.12·19-s − 2.95·20-s + 3.15·22-s − 7.95·23-s + 3.72·25-s − 0.00112·26-s + 0.794·28-s − 2.72·29-s + 7.54·31-s + 32-s − 4.32·34-s − 2.34·35-s + 6.76·37-s + 3.12·38-s − 2.95·40-s − 11.8·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.32·5-s + 0.300·7-s + 0.353·8-s − 0.934·10-s + 0.950·11-s − 0.000311·13-s + 0.212·14-s + 0.250·16-s − 1.04·17-s + 0.716·19-s − 0.660·20-s + 0.671·22-s − 1.65·23-s + 0.745·25-s − 0.000220·26-s + 0.150·28-s − 0.506·29-s + 1.35·31-s + 0.176·32-s − 0.741·34-s − 0.396·35-s + 1.11·37-s + 0.506·38-s − 0.467·40-s − 1.84·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 149 | \( 1 - T \) |
| good | 5 | \( 1 + 2.95T + 5T^{2} \) |
| 7 | \( 1 - 0.794T + 7T^{2} \) |
| 11 | \( 1 - 3.15T + 11T^{2} \) |
| 13 | \( 1 + 0.00112T + 13T^{2} \) |
| 17 | \( 1 + 4.32T + 17T^{2} \) |
| 19 | \( 1 - 3.12T + 19T^{2} \) |
| 23 | \( 1 + 7.95T + 23T^{2} \) |
| 29 | \( 1 + 2.72T + 29T^{2} \) |
| 31 | \( 1 - 7.54T + 31T^{2} \) |
| 37 | \( 1 - 6.76T + 37T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 + 4.75T + 43T^{2} \) |
| 47 | \( 1 + 5.13T + 47T^{2} \) |
| 53 | \( 1 - 2.65T + 53T^{2} \) |
| 59 | \( 1 - 8.78T + 59T^{2} \) |
| 61 | \( 1 + 4.01T + 61T^{2} \) |
| 67 | \( 1 + 8.23T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 - 11.1T + 73T^{2} \) |
| 79 | \( 1 + 5.69T + 79T^{2} \) |
| 83 | \( 1 - 11.5T + 83T^{2} \) |
| 89 | \( 1 + 6.40T + 89T^{2} \) |
| 97 | \( 1 + 16.6T + 97T^{2} \) |
| 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
−7.46436058497550566156324436832, −6.66135417695932616139592507445, −6.26824184966806563511956030516, −5.18095525502081341142731644130, −4.52623256168734195072356213641, −3.92329713565858156288138658239, −3.42689845472933731685331142076, −2.35623822558099187481601478442, −1.35213795137843744330786339217, 0,
1.35213795137843744330786339217, 2.35623822558099187481601478442, 3.42689845472933731685331142076, 3.92329713565858156288138658239, 4.52623256168734195072356213641, 5.18095525502081341142731644130, 6.26824184966806563511956030516, 6.66135417695932616139592507445, 7.46436058497550566156324436832
