| L(s) = 1 | + 2-s + 0.436·3-s + 4-s − 0.0518·5-s + 0.436·6-s + 0.167·7-s + 8-s − 2.80·9-s − 0.0518·10-s + 1.44·11-s + 0.436·12-s + 4.01·13-s + 0.167·14-s − 0.0226·15-s + 16-s − 2.41·17-s − 2.80·18-s − 2.67·19-s − 0.0518·20-s + 0.0730·21-s + 1.44·22-s − 6.72·23-s + 0.436·24-s − 4.99·25-s + 4.01·26-s − 2.53·27-s + 0.167·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.252·3-s + 0.5·4-s − 0.0231·5-s + 0.178·6-s + 0.0632·7-s + 0.353·8-s − 0.936·9-s − 0.0163·10-s + 0.435·11-s + 0.126·12-s + 1.11·13-s + 0.0447·14-s − 0.00584·15-s + 0.250·16-s − 0.584·17-s − 0.662·18-s − 0.614·19-s − 0.0115·20-s + 0.0159·21-s + 0.307·22-s − 1.40·23-s + 0.0891·24-s − 0.999·25-s + 0.787·26-s − 0.488·27-s + 0.0316·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| 97 | \( 1 + T \) |
| good | 3 | \( 1 - 0.436T + 3T^{2} \) |
| 5 | \( 1 + 0.0518T + 5T^{2} \) |
| 7 | \( 1 - 0.167T + 7T^{2} \) |
| 11 | \( 1 - 1.44T + 11T^{2} \) |
| 13 | \( 1 - 4.01T + 13T^{2} \) |
| 17 | \( 1 + 2.41T + 17T^{2} \) |
| 19 | \( 1 + 2.67T + 19T^{2} \) |
| 23 | \( 1 + 6.72T + 23T^{2} \) |
| 29 | \( 1 + 8.89T + 29T^{2} \) |
| 37 | \( 1 + 7.01T + 37T^{2} \) |
| 41 | \( 1 + 2.01T + 41T^{2} \) |
| 43 | \( 1 - 2.30T + 43T^{2} \) |
| 47 | \( 1 - 2.67T + 47T^{2} \) |
| 53 | \( 1 + 7.23T + 53T^{2} \) |
| 59 | \( 1 - 8.40T + 59T^{2} \) |
| 61 | \( 1 - 2.45T + 61T^{2} \) |
| 67 | \( 1 + 12.3T + 67T^{2} \) |
| 71 | \( 1 + 3.10T + 71T^{2} \) |
| 73 | \( 1 - 5.71T + 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 - 8.37T + 83T^{2} \) |
| 89 | \( 1 - 6.86T + 89T^{2} \) |
| 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
−7.81596004871048914953821701747, −6.84611381006243459827464219904, −6.07283029217639625175179068554, −5.77916495842283343705724112694, −4.77888574490762223839377648105, −3.77824761874631445788328600303, −3.57979937506029668428448362945, −2.32294418438251007107687247411, −1.69084102594602098137382313825, 0,
1.69084102594602098137382313825, 2.32294418438251007107687247411, 3.57979937506029668428448362945, 3.77824761874631445788328600303, 4.77888574490762223839377648105, 5.77916495842283343705724112694, 6.07283029217639625175179068554, 6.84611381006243459827464219904, 7.81596004871048914953821701747
