| L(s) = 1 | − 1.29·2-s + 0.182·3-s − 0.331·4-s − 5-s − 0.236·6-s + 2.69·7-s + 3.01·8-s − 2.96·9-s + 1.29·10-s − 0.674·11-s − 0.0606·12-s + 0.890·13-s − 3.48·14-s − 0.182·15-s − 3.22·16-s − 5.74·17-s + 3.83·18-s + 7.13·19-s + 0.331·20-s + 0.492·21-s + 0.870·22-s − 5.38·23-s + 0.550·24-s + 25-s − 1.15·26-s − 1.09·27-s − 0.893·28-s + ⋯ |
| L(s) = 1 | − 0.913·2-s + 0.105·3-s − 0.165·4-s − 0.447·5-s − 0.0964·6-s + 1.01·7-s + 1.06·8-s − 0.988·9-s + 0.408·10-s − 0.203·11-s − 0.0175·12-s + 0.246·13-s − 0.930·14-s − 0.0472·15-s − 0.806·16-s − 1.39·17-s + 0.903·18-s + 1.63·19-s + 0.0741·20-s + 0.107·21-s + 0.185·22-s − 1.12·23-s + 0.112·24-s + 0.200·25-s − 0.225·26-s − 0.210·27-s − 0.168·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 + 1.29T + 2T^{2} \) |
| 3 | \( 1 - 0.182T + 3T^{2} \) |
| 7 | \( 1 - 2.69T + 7T^{2} \) |
| 11 | \( 1 + 0.674T + 11T^{2} \) |
| 13 | \( 1 - 0.890T + 13T^{2} \) |
| 17 | \( 1 + 5.74T + 17T^{2} \) |
| 19 | \( 1 - 7.13T + 19T^{2} \) |
| 23 | \( 1 + 5.38T + 23T^{2} \) |
| 29 | \( 1 - 1.73T + 29T^{2} \) |
| 31 | \( 1 - 1.10T + 31T^{2} \) |
| 37 | \( 1 - 9.69T + 37T^{2} \) |
| 41 | \( 1 + 6.03T + 41T^{2} \) |
| 43 | \( 1 + 12.0T + 43T^{2} \) |
| 47 | \( 1 - 3.21T + 47T^{2} \) |
| 53 | \( 1 + 1.42T + 53T^{2} \) |
| 59 | \( 1 - 10.5T + 59T^{2} \) |
| 61 | \( 1 + 5.44T + 61T^{2} \) |
| 67 | \( 1 - 14.8T + 67T^{2} \) |
| 71 | \( 1 + 15.7T + 71T^{2} \) |
| 73 | \( 1 + 5.85T + 73T^{2} \) |
| 79 | \( 1 - 3.02T + 79T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 + 5.59T + 89T^{2} \) |
| 97 | \( 1 + 17.6T + 97T^{2} \) |
| 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
−8.549187130751554166298468313941, −8.215587745401088854463906353202, −7.60461296325844575277688464694, −6.61299820668580002960928789194, −5.42269135453949747538861899744, −4.73840972388235015762898202175, −3.82759227272561915082558616976, −2.55352252193604629084116675832, −1.36333658457255542948809081595, 0,
1.36333658457255542948809081595, 2.55352252193604629084116675832, 3.82759227272561915082558616976, 4.73840972388235015762898202175, 5.42269135453949747538861899744, 6.61299820668580002960928789194, 7.60461296325844575277688464694, 8.215587745401088854463906353202, 8.549187130751554166298468313941
