| L(s) = 1 | + 2-s − 2.64·3-s + 4-s − 3.78·5-s − 2.64·6-s − 4.25·7-s + 8-s + 4.01·9-s − 3.78·10-s − 5.82·11-s − 2.64·12-s + 6.86·13-s − 4.25·14-s + 10.0·15-s + 16-s + 1.86·17-s + 4.01·18-s + 4.22·19-s − 3.78·20-s + 11.2·21-s − 5.82·22-s − 0.209·23-s − 2.64·24-s + 9.29·25-s + 6.86·26-s − 2.68·27-s − 4.25·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.52·3-s + 0.5·4-s − 1.69·5-s − 1.08·6-s − 1.60·7-s + 0.353·8-s + 1.33·9-s − 1.19·10-s − 1.75·11-s − 0.764·12-s + 1.90·13-s − 1.13·14-s + 2.58·15-s + 0.250·16-s + 0.453·17-s + 0.946·18-s + 0.968·19-s − 0.845·20-s + 2.45·21-s − 1.24·22-s − 0.0437·23-s − 0.540·24-s + 1.85·25-s + 1.34·26-s − 0.516·27-s − 0.803·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 2003 | \( 1 - T \) |
| good | 3 | \( 1 + 2.64T + 3T^{2} \) |
| 5 | \( 1 + 3.78T + 5T^{2} \) |
| 7 | \( 1 + 4.25T + 7T^{2} \) |
| 11 | \( 1 + 5.82T + 11T^{2} \) |
| 13 | \( 1 - 6.86T + 13T^{2} \) |
| 17 | \( 1 - 1.86T + 17T^{2} \) |
| 19 | \( 1 - 4.22T + 19T^{2} \) |
| 23 | \( 1 + 0.209T + 23T^{2} \) |
| 29 | \( 1 - 3.68T + 29T^{2} \) |
| 31 | \( 1 - 8.12T + 31T^{2} \) |
| 37 | \( 1 + 1.63T + 37T^{2} \) |
| 41 | \( 1 + 8.60T + 41T^{2} \) |
| 43 | \( 1 + 9.87T + 43T^{2} \) |
| 47 | \( 1 + 8.62T + 47T^{2} \) |
| 53 | \( 1 - 8.31T + 53T^{2} \) |
| 59 | \( 1 + 1.12T + 59T^{2} \) |
| 61 | \( 1 + 2.39T + 61T^{2} \) |
| 67 | \( 1 - 10.6T + 67T^{2} \) |
| 71 | \( 1 + 2.94T + 71T^{2} \) |
| 73 | \( 1 - 7.66T + 73T^{2} \) |
| 79 | \( 1 - 1.26T + 79T^{2} \) |
| 83 | \( 1 + 10.6T + 83T^{2} \) |
| 89 | \( 1 + 13.6T + 89T^{2} \) |
| 97 | \( 1 - 7.16T + 97T^{2} \) |
| 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
−8.020122155208443982227060439516, −6.92017372769177659661836441114, −6.63399649607795128390835178520, −5.75949154607139464371198108738, −5.17548879067621865280788858421, −4.35111946147885766022562368892, −3.35694088773145669128504455308, −3.12084847336680051547867185279, −0.932861187716976259342304613509, 0,
0.932861187716976259342304613509, 3.12084847336680051547867185279, 3.35694088773145669128504455308, 4.35111946147885766022562368892, 5.17548879067621865280788858421, 5.75949154607139464371198108738, 6.63399649607795128390835178520, 6.92017372769177659661836441114, 8.020122155208443982227060439516
