| L(s) = 1 | + 2-s + 3-s + 4-s − 3.87·5-s + 6-s + 1.22·7-s + 8-s + 9-s − 3.87·10-s − 2.12·11-s + 12-s + 0.490·13-s + 1.22·14-s − 3.87·15-s + 16-s − 5.53·17-s + 18-s − 3.87·20-s + 1.22·21-s − 2.12·22-s − 8.94·23-s + 24-s + 10.0·25-s + 0.490·26-s + 27-s + 1.22·28-s − 8.47·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.73·5-s + 0.408·6-s + 0.463·7-s + 0.353·8-s + 0.333·9-s − 1.22·10-s − 0.639·11-s + 0.288·12-s + 0.135·13-s + 0.327·14-s − 1.00·15-s + 0.250·16-s − 1.34·17-s + 0.235·18-s − 0.867·20-s + 0.267·21-s − 0.452·22-s − 1.86·23-s + 0.204·24-s + 2.00·25-s + 0.0961·26-s + 0.192·27-s + 0.231·28-s − 1.57·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 + 3.87T + 5T^{2} \) |
| 7 | \( 1 - 1.22T + 7T^{2} \) |
| 11 | \( 1 + 2.12T + 11T^{2} \) |
| 13 | \( 1 - 0.490T + 13T^{2} \) |
| 17 | \( 1 + 5.53T + 17T^{2} \) |
| 23 | \( 1 + 8.94T + 23T^{2} \) |
| 29 | \( 1 + 8.47T + 29T^{2} \) |
| 31 | \( 1 - 2.41T + 31T^{2} \) |
| 37 | \( 1 - 1.69T + 37T^{2} \) |
| 41 | \( 1 - 1.59T + 41T^{2} \) |
| 43 | \( 1 + 6.63T + 43T^{2} \) |
| 47 | \( 1 - 2.17T + 47T^{2} \) |
| 53 | \( 1 - 8.88T + 53T^{2} \) |
| 59 | \( 1 + 11.4T + 59T^{2} \) |
| 61 | \( 1 - 0.0418T + 61T^{2} \) |
| 67 | \( 1 - 4.47T + 67T^{2} \) |
| 71 | \( 1 - 2.63T + 71T^{2} \) |
| 73 | \( 1 + 15.3T + 73T^{2} \) |
| 79 | \( 1 + 4.66T + 79T^{2} \) |
| 83 | \( 1 - 12.4T + 83T^{2} \) |
| 89 | \( 1 - 8.45T + 89T^{2} \) |
| 97 | \( 1 + 14.3T + 97T^{2} \) |
| 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
−8.325429143533674064325373990886, −7.933817003779015190342910496849, −7.30461466378993975813350729831, −6.41648168573459790601606815175, −5.26230852022658404317002565540, −4.27641407707610209618361783596, −3.98129765993333304517105581890, −2.95701712481061046298557494549, −1.90862217838124315608554626132, 0,
1.90862217838124315608554626132, 2.95701712481061046298557494549, 3.98129765993333304517105581890, 4.27641407707610209618361783596, 5.26230852022658404317002565540, 6.41648168573459790601606815175, 7.30461466378993975813350729831, 7.933817003779015190342910496849, 8.325429143533674064325373990886
