| L(s) = 1 | + 1.39·2-s + 3-s − 0.0658·4-s − 1.39·5-s + 1.39·6-s − 4.67·7-s − 2.87·8-s + 9-s − 1.93·10-s + 2.15·11-s − 0.0658·12-s + 3.57·13-s − 6.50·14-s − 1.39·15-s − 3.86·16-s + 17-s + 1.39·18-s + 4.57·19-s + 0.0918·20-s − 4.67·21-s + 2.99·22-s + 5.74·23-s − 2.87·24-s − 3.05·25-s + 4.96·26-s + 27-s + 0.307·28-s + ⋯ |
| L(s) = 1 | + 0.983·2-s + 0.577·3-s − 0.0329·4-s − 0.623·5-s + 0.567·6-s − 1.76·7-s − 1.01·8-s + 0.333·9-s − 0.613·10-s + 0.649·11-s − 0.0190·12-s + 0.990·13-s − 1.73·14-s − 0.360·15-s − 0.966·16-s + 0.242·17-s + 0.327·18-s + 1.05·19-s + 0.0205·20-s − 1.02·21-s + 0.638·22-s + 1.19·23-s − 0.586·24-s − 0.610·25-s + 0.973·26-s + 0.192·27-s + 0.0581·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 157 | \( 1 + T \) |
| good | 2 | \( 1 - 1.39T + 2T^{2} \) |
| 5 | \( 1 + 1.39T + 5T^{2} \) |
| 7 | \( 1 + 4.67T + 7T^{2} \) |
| 11 | \( 1 - 2.15T + 11T^{2} \) |
| 13 | \( 1 - 3.57T + 13T^{2} \) |
| 19 | \( 1 - 4.57T + 19T^{2} \) |
| 23 | \( 1 - 5.74T + 23T^{2} \) |
| 29 | \( 1 - 4.63T + 29T^{2} \) |
| 31 | \( 1 + 1.78T + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 41 | \( 1 - 9.38T + 41T^{2} \) |
| 43 | \( 1 + 6.68T + 43T^{2} \) |
| 47 | \( 1 + 10.1T + 47T^{2} \) |
| 53 | \( 1 - 1.21T + 53T^{2} \) |
| 59 | \( 1 + 1.72T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 + 13.3T + 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 + 2.14T + 73T^{2} \) |
| 79 | \( 1 + 7.86T + 79T^{2} \) |
| 83 | \( 1 - 0.486T + 83T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 + 14.7T + 97T^{2} \) |
| 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
−7.26837029265912815455623758474, −6.68203592499384820856112878768, −6.11070738129826119120973242940, −5.37278034149903366132815694839, −4.44021218911943829871495667752, −3.71075509534727738024886947090, −3.29294933855890790712932246434, −2.88339945180212485711523476046, −1.27086515514899664308103849069, 0,
1.27086515514899664308103849069, 2.88339945180212485711523476046, 3.29294933855890790712932246434, 3.71075509534727738024886947090, 4.44021218911943829871495667752, 5.37278034149903366132815694839, 6.11070738129826119120973242940, 6.68203592499384820856112878768, 7.26837029265912815455623758474
