| L(s) = 1 | + 2-s + 0.618·3-s + 4-s − 5-s + 0.618·6-s − 7-s + 8-s − 2.61·9-s − 10-s + 0.618·12-s + 5.23·13-s − 14-s − 0.618·15-s + 16-s + 1.61·17-s − 2.61·18-s + 0.381·19-s − 20-s − 0.618·21-s − 8.47·23-s + 0.618·24-s + 25-s + 5.23·26-s − 3.47·27-s − 28-s − 4·29-s − 0.618·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.356·3-s + 0.5·4-s − 0.447·5-s + 0.252·6-s − 0.377·7-s + 0.353·8-s − 0.872·9-s − 0.316·10-s + 0.178·12-s + 1.45·13-s − 0.267·14-s − 0.159·15-s + 0.250·16-s + 0.392·17-s − 0.617·18-s + 0.0876·19-s − 0.223·20-s − 0.134·21-s − 1.76·23-s + 0.126·24-s + 0.200·25-s + 1.02·26-s − 0.668·27-s − 0.188·28-s − 0.742·29-s − 0.112·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 0.618T + 3T^{2} \) |
| 13 | \( 1 - 5.23T + 13T^{2} \) |
| 17 | \( 1 - 1.61T + 17T^{2} \) |
| 19 | \( 1 - 0.381T + 19T^{2} \) |
| 23 | \( 1 + 8.47T + 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 + 4.47T + 31T^{2} \) |
| 37 | \( 1 - 9.70T + 37T^{2} \) |
| 41 | \( 1 + 8.61T + 41T^{2} \) |
| 43 | \( 1 - 4.61T + 43T^{2} \) |
| 47 | \( 1 + 6.76T + 47T^{2} \) |
| 53 | \( 1 + 6.47T + 53T^{2} \) |
| 59 | \( 1 + 3.61T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 + 6.09T + 67T^{2} \) |
| 71 | \( 1 - 2.47T + 71T^{2} \) |
| 73 | \( 1 + 4.32T + 73T^{2} \) |
| 79 | \( 1 + 1.70T + 79T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 - 9.85T + 89T^{2} \) |
| 97 | \( 1 + 10.7T + 97T^{2} \) |
| 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
−7.53712659283738338634522538154, −6.55479876292262841770399356059, −5.96207261299810892494366380299, −5.54832036920893135520857945650, −4.46062034475535623420946684046, −3.69207901477114918765426925616, −3.36111319641238744756435358516, −2.42194252831615710031511039339, −1.44556354007208217400048165626, 0,
1.44556354007208217400048165626, 2.42194252831615710031511039339, 3.36111319641238744756435358516, 3.69207901477114918765426925616, 4.46062034475535623420946684046, 5.54832036920893135520857945650, 5.96207261299810892494366380299, 6.55479876292262841770399356059, 7.53712659283738338634522538154
