| L(s) = 1 | − 2-s + 2.02·3-s + 4-s − 3.63·5-s − 2.02·6-s + 2.67·7-s − 8-s + 1.09·9-s + 3.63·10-s + 1.35·11-s + 2.02·12-s + 0.698·13-s − 2.67·14-s − 7.35·15-s + 16-s − 2.67·17-s − 1.09·18-s + 5.59·19-s − 3.63·20-s + 5.41·21-s − 1.35·22-s − 8.27·23-s − 2.02·24-s + 8.21·25-s − 0.698·26-s − 3.86·27-s + 2.67·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.16·3-s + 0.5·4-s − 1.62·5-s − 0.825·6-s + 1.01·7-s − 0.353·8-s + 0.363·9-s + 1.14·10-s + 0.407·11-s + 0.583·12-s + 0.193·13-s − 0.715·14-s − 1.89·15-s + 0.250·16-s − 0.648·17-s − 0.257·18-s + 1.28·19-s − 0.812·20-s + 1.18·21-s − 0.287·22-s − 1.72·23-s − 0.412·24-s + 1.64·25-s − 0.136·26-s − 0.743·27-s + 0.505·28-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 | \( 1 - 2.02T + 3T^{2} \) |
| 5 | \( 1 + 3.63T + 5T^{2} \) |
| 7 | \( 1 - 2.67T + 7T^{2} \) |
| 11 | \( 1 - 1.35T + 11T^{2} \) |
| 13 | \( 1 - 0.698T + 13T^{2} \) |
| 17 | \( 1 + 2.67T + 17T^{2} \) |
| 19 | \( 1 - 5.59T + 19T^{2} \) |
| 23 | \( 1 + 8.27T + 23T^{2} \) |
| 29 | \( 1 + 4.22T + 29T^{2} \) |
| 31 | \( 1 + 5.29T + 31T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 + 0.202T + 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 - 5.05T + 47T^{2} \) |
| 53 | \( 1 + 9.65T + 53T^{2} \) |
| 59 | \( 1 - 3.66T + 59T^{2} \) |
| 61 | \( 1 - 0.421T + 61T^{2} \) |
| 67 | \( 1 - 7.87T + 67T^{2} \) |
| 71 | \( 1 + 5.64T + 71T^{2} \) |
| 73 | \( 1 + 8.92T + 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 - 9.01T + 89T^{2} \) |
| 97 | \( 1 + 5.97T + 97T^{2} \) |
| 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
−8.078259142341430260810730613620, −7.55367286783786034878875339749, −7.26373196107643254782904630484, −5.97065875613523946621980815479, −4.89488094497907759815953064887, −3.83677876875278966650060777432, −3.57862309334043812214895923577, −2.38378663548664788009525061038, −1.47644253471436725255526731398, 0,
1.47644253471436725255526731398, 2.38378663548664788009525061038, 3.57862309334043812214895923577, 3.83677876875278966650060777432, 4.89488094497907759815953064887, 5.97065875613523946621980815479, 7.26373196107643254782904630484, 7.55367286783786034878875339749, 8.078259142341430260810730613620
