| L(s) = 1 | − 2-s − 0.129·3-s + 4-s + 1.90·5-s + 0.129·6-s − 0.0260·7-s − 8-s − 2.98·9-s − 1.90·10-s + 2.55·11-s − 0.129·12-s − 4.06·13-s + 0.0260·14-s − 0.247·15-s + 16-s − 4.11·17-s + 2.98·18-s + 19-s + 1.90·20-s + 0.00338·21-s − 2.55·22-s + 3.49·23-s + 0.129·24-s − 1.36·25-s + 4.06·26-s + 0.776·27-s − 0.0260·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.0749·3-s + 0.5·4-s + 0.852·5-s + 0.0530·6-s − 0.00983·7-s − 0.353·8-s − 0.994·9-s − 0.602·10-s + 0.771·11-s − 0.0374·12-s − 1.12·13-s + 0.00695·14-s − 0.0639·15-s + 0.250·16-s − 0.998·17-s + 0.703·18-s + 0.229·19-s + 0.426·20-s + 0.000737·21-s − 0.545·22-s + 0.727·23-s + 0.0265·24-s − 0.273·25-s + 0.796·26-s + 0.149·27-s − 0.00491·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 - T \) |
| good | 3 | \( 1 + 0.129T + 3T^{2} \) |
| 5 | \( 1 - 1.90T + 5T^{2} \) |
| 7 | \( 1 + 0.0260T + 7T^{2} \) |
| 11 | \( 1 - 2.55T + 11T^{2} \) |
| 13 | \( 1 + 4.06T + 13T^{2} \) |
| 17 | \( 1 + 4.11T + 17T^{2} \) |
| 23 | \( 1 - 3.49T + 23T^{2} \) |
| 29 | \( 1 + 4.54T + 29T^{2} \) |
| 31 | \( 1 - 4.14T + 31T^{2} \) |
| 37 | \( 1 - 9.52T + 37T^{2} \) |
| 41 | \( 1 - 6.26T + 41T^{2} \) |
| 43 | \( 1 + 3.53T + 43T^{2} \) |
| 47 | \( 1 - 4.46T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 - 3.30T + 59T^{2} \) |
| 61 | \( 1 - 5.56T + 61T^{2} \) |
| 67 | \( 1 - 3.69T + 67T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 73 | \( 1 + 6.37T + 73T^{2} \) |
| 79 | \( 1 - 17.2T + 79T^{2} \) |
| 83 | \( 1 - 5.79T + 83T^{2} \) |
| 89 | \( 1 + 16.4T + 89T^{2} \) |
| 97 | \( 1 + 16.7T + 97T^{2} \) |
| 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
−7.61762885008590012907036278144, −6.63603061730958681133469526797, −6.34486717334009484595047059446, −5.49647580759479077677022296978, −4.84544694129897118157989135556, −3.82873044885255873637245117532, −2.69156580610313144546807405380, −2.29663862181151810522681007240, −1.20528802362214142122963733857, 0,
1.20528802362214142122963733857, 2.29663862181151810522681007240, 2.69156580610313144546807405380, 3.82873044885255873637245117532, 4.84544694129897118157989135556, 5.49647580759479077677022296978, 6.34486717334009484595047059446, 6.63603061730958681133469526797, 7.61762885008590012907036278144
