| L(s) = 1 | + 2.13·2-s − 2.37·3-s + 2.55·4-s + 5-s − 5.06·6-s + 7-s + 1.17·8-s + 2.64·9-s + 2.13·10-s − 2.50·11-s − 6.06·12-s + 2.58·13-s + 2.13·14-s − 2.37·15-s − 2.59·16-s − 2.98·17-s + 5.64·18-s + 1.94·19-s + 2.55·20-s − 2.37·21-s − 5.34·22-s + 1.04·23-s − 2.79·24-s + 25-s + 5.51·26-s + 0.838·27-s + 2.55·28-s + ⋯ |
| L(s) = 1 | + 1.50·2-s − 1.37·3-s + 1.27·4-s + 0.447·5-s − 2.06·6-s + 0.377·7-s + 0.415·8-s + 0.882·9-s + 0.674·10-s − 0.755·11-s − 1.75·12-s + 0.717·13-s + 0.570·14-s − 0.613·15-s − 0.648·16-s − 0.723·17-s + 1.33·18-s + 0.446·19-s + 0.570·20-s − 0.518·21-s − 1.13·22-s + 0.216·23-s − 0.570·24-s + 0.200·25-s + 1.08·26-s + 0.161·27-s + 0.482·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 229 | \( 1 + T \) |
| good | 2 | \( 1 - 2.13T + 2T^{2} \) |
| 3 | \( 1 + 2.37T + 3T^{2} \) |
| 11 | \( 1 + 2.50T + 11T^{2} \) |
| 13 | \( 1 - 2.58T + 13T^{2} \) |
| 17 | \( 1 + 2.98T + 17T^{2} \) |
| 19 | \( 1 - 1.94T + 19T^{2} \) |
| 23 | \( 1 - 1.04T + 23T^{2} \) |
| 29 | \( 1 - 3.02T + 29T^{2} \) |
| 31 | \( 1 + 7.49T + 31T^{2} \) |
| 37 | \( 1 - 3.57T + 37T^{2} \) |
| 41 | \( 1 + 1.89T + 41T^{2} \) |
| 43 | \( 1 + 2.55T + 43T^{2} \) |
| 47 | \( 1 + 6.29T + 47T^{2} \) |
| 53 | \( 1 + 9.62T + 53T^{2} \) |
| 59 | \( 1 - 12.9T + 59T^{2} \) |
| 61 | \( 1 + 13.3T + 61T^{2} \) |
| 67 | \( 1 - 4.52T + 67T^{2} \) |
| 71 | \( 1 - 2.58T + 71T^{2} \) |
| 73 | \( 1 - 4.64T + 73T^{2} \) |
| 79 | \( 1 + 0.392T + 79T^{2} \) |
| 83 | \( 1 + 8.51T + 83T^{2} \) |
| 89 | \( 1 - 8.44T + 89T^{2} \) |
| 97 | \( 1 + 6.56T + 97T^{2} \) |
| 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
−6.99492020110284584553136343414, −6.45655979238363716382685061810, −5.91686047037648919207820737245, −5.23586098576635146658113369240, −4.97462455758085705065304730782, −4.18571898093284925243806235544, −3.30113088181923908208735758007, −2.42097363026113793995123031530, −1.38940230921731347667398411043, 0,
1.38940230921731347667398411043, 2.42097363026113793995123031530, 3.30113088181923908208735758007, 4.18571898093284925243806235544, 4.97462455758085705065304730782, 5.23586098576635146658113369240, 5.91686047037648919207820737245, 6.45655979238363716382685061810, 6.99492020110284584553136343414
