L(s) = 1 | + 2-s − 3-s + 4-s − 2·5-s − 6-s + 7-s + 8-s + 9-s − 2·10-s − 12-s − 4·13-s + 14-s + 2·15-s + 16-s + 18-s − 6·19-s − 2·20-s − 21-s − 23-s − 24-s − 25-s − 4·26-s − 27-s + 28-s + 6·29-s + 2·30-s − 10·31-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.632·10-s − 0.288·12-s − 1.10·13-s + 0.267·14-s + 0.516·15-s + 1/4·16-s + 0.235·18-s − 1.37·19-s − 0.447·20-s − 0.218·21-s − 0.208·23-s − 0.204·24-s − 1/5·25-s − 0.784·26-s − 0.192·27-s + 0.188·28-s + 1.11·29-s + 0.365·30-s − 1.79·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.09025405151783, −13.60044679400944, −12.92992586818529, −12.53490611763396, −12.17623564432026, −11.75226538548475, −11.29326296601170, −10.83747210434714, −10.25003393837036, −10.02224707104492, −9.022921360429891, −8.686697628736699, −7.972834855445565, −7.505632778577528, −7.127788807915027, −6.563034953138422, −5.987719605102971, −5.419646894074982, −4.852848180985605, −4.429765870079360, −4.002155539587804, −3.339821829693753, −2.666520689027754, −1.937322949478573, −1.388478174788784, 0, 0,
1.388478174788784, 1.937322949478573, 2.666520689027754, 3.339821829693753, 4.002155539587804, 4.429765870079360, 4.852848180985605, 5.419646894074982, 5.987719605102971, 6.563034953138422, 7.127788807915027, 7.505632778577528, 7.972834855445565, 8.686697628736699, 9.022921360429891, 10.02224707104492, 10.25003393837036, 10.83747210434714, 11.29326296601170, 11.75226538548475, 12.17623564432026, 12.53490611763396, 12.92992586818529, 13.60044679400944, 14.09025405151783