| L(s) = 1 | − 0.693·2-s − 3-s − 1.51·4-s − 5-s + 0.693·6-s − 0.747·7-s + 2.44·8-s + 9-s + 0.693·10-s − 0.228·11-s + 1.51·12-s + 0.572·13-s + 0.518·14-s + 15-s + 1.34·16-s + 5.57·17-s − 0.693·18-s + 2.97·19-s + 1.51·20-s + 0.747·21-s + 0.158·22-s − 5.02·23-s − 2.44·24-s + 25-s − 0.397·26-s − 27-s + 1.13·28-s + ⋯ |
| L(s) = 1 | − 0.490·2-s − 0.577·3-s − 0.759·4-s − 0.447·5-s + 0.283·6-s − 0.282·7-s + 0.863·8-s + 0.333·9-s + 0.219·10-s − 0.0690·11-s + 0.438·12-s + 0.158·13-s + 0.138·14-s + 0.258·15-s + 0.335·16-s + 1.35·17-s − 0.163·18-s + 0.682·19-s + 0.339·20-s + 0.163·21-s + 0.0338·22-s − 1.04·23-s − 0.498·24-s + 0.200·25-s − 0.0778·26-s − 0.192·27-s + 0.214·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 + T \) |
| good | 2 | \( 1 + 0.693T + 2T^{2} \) |
| 7 | \( 1 + 0.747T + 7T^{2} \) |
| 11 | \( 1 + 0.228T + 11T^{2} \) |
| 13 | \( 1 - 0.572T + 13T^{2} \) |
| 17 | \( 1 - 5.57T + 17T^{2} \) |
| 19 | \( 1 - 2.97T + 19T^{2} \) |
| 23 | \( 1 + 5.02T + 23T^{2} \) |
| 29 | \( 1 + 2.93T + 29T^{2} \) |
| 31 | \( 1 + 8.58T + 31T^{2} \) |
| 37 | \( 1 - 6.76T + 37T^{2} \) |
| 41 | \( 1 - 2.31T + 41T^{2} \) |
| 43 | \( 1 + 11.4T + 43T^{2} \) |
| 47 | \( 1 - 4.84T + 47T^{2} \) |
| 53 | \( 1 - 6.62T + 53T^{2} \) |
| 59 | \( 1 + 2.19T + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 71 | \( 1 - 5.77T + 71T^{2} \) |
| 73 | \( 1 + 13.3T + 73T^{2} \) |
| 79 | \( 1 + 8.65T + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 + 8.65T + 89T^{2} \) |
| 97 | \( 1 - 11.3T + 97T^{2} \) |
| 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
−9.688688137579021572112932066279, −8.785415445543456119183861552547, −7.81066064107239263359691323115, −7.34024844933302578443113714538, −5.97941159575107461235336895158, −5.27211035871461031560448375250, −4.20959850272418837534530150717, −3.34618615445345844967684480489, −1.40058979067665759263629680093, 0,
1.40058979067665759263629680093, 3.34618615445345844967684480489, 4.20959850272418837534530150717, 5.27211035871461031560448375250, 5.97941159575107461235336895158, 7.34024844933302578443113714538, 7.81066064107239263359691323115, 8.785415445543456119183861552547, 9.688688137579021572112932066279
