| L(s) = 1 | − 2-s + 3-s + 4-s − 2·5-s − 6-s − 2·7-s − 8-s + 9-s + 2·10-s + 12-s + 2·14-s − 2·15-s + 16-s + 2·17-s − 18-s + 6·19-s − 2·20-s − 2·21-s − 4·23-s − 24-s − 25-s + 27-s − 2·28-s − 10·29-s + 2·30-s − 10·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s + 0.288·12-s + 0.534·14-s − 0.516·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 1.37·19-s − 0.447·20-s − 0.436·21-s − 0.834·23-s − 0.204·24-s − 1/5·25-s + 0.192·27-s − 0.377·28-s − 1.85·29-s + 0.365·30-s − 1.79·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
| 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
−9.640791322379534312317268263332, −8.733226088694874330202531558278, −7.72635236122700537791758242828, −7.48225643732914189291632299460, −6.36202848183618365798599646203, −5.25884034922839215249485111848, −3.72725018247659066479821371636, −3.27299015443475162705655779949, −1.76712137722831607625576187625, 0,
1.76712137722831607625576187625, 3.27299015443475162705655779949, 3.72725018247659066479821371636, 5.25884034922839215249485111848, 6.36202848183618365798599646203, 7.48225643732914189291632299460, 7.72635236122700537791758242828, 8.733226088694874330202531558278, 9.640791322379534312317268263332
