| L(s) = 1 | + 2-s − 2.02·3-s + 4-s + 2.47·5-s − 2.02·6-s + 1.41·7-s + 8-s + 1.11·9-s + 2.47·10-s − 1.90·11-s − 2.02·12-s + 3.04·13-s + 1.41·14-s − 5.01·15-s + 16-s − 0.777·17-s + 1.11·18-s + 19-s + 2.47·20-s − 2.86·21-s − 1.90·22-s − 7.98·23-s − 2.02·24-s + 1.10·25-s + 3.04·26-s + 3.82·27-s + 1.41·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.17·3-s + 0.5·4-s + 1.10·5-s − 0.828·6-s + 0.534·7-s + 0.353·8-s + 0.371·9-s + 0.781·10-s − 0.573·11-s − 0.585·12-s + 0.845·13-s + 0.378·14-s − 1.29·15-s + 0.250·16-s − 0.188·17-s + 0.262·18-s + 0.229·19-s + 0.552·20-s − 0.626·21-s − 0.405·22-s − 1.66·23-s − 0.414·24-s + 0.221·25-s + 0.598·26-s + 0.736·27-s + 0.267·28-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 | \( 1 + 2.02T + 3T^{2} \) |
| 5 | \( 1 - 2.47T + 5T^{2} \) |
| 7 | \( 1 - 1.41T + 7T^{2} \) |
| 11 | \( 1 + 1.90T + 11T^{2} \) |
| 13 | \( 1 - 3.04T + 13T^{2} \) |
| 17 | \( 1 + 0.777T + 17T^{2} \) |
| 23 | \( 1 + 7.98T + 23T^{2} \) |
| 29 | \( 1 + 5.89T + 29T^{2} \) |
| 31 | \( 1 + 5.52T + 31T^{2} \) |
| 37 | \( 1 + 3.09T + 37T^{2} \) |
| 41 | \( 1 + 6.61T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 + 6.04T + 47T^{2} \) |
| 53 | \( 1 + 3.46T + 53T^{2} \) |
| 59 | \( 1 + 1.73T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 - 5.11T + 67T^{2} \) |
| 71 | \( 1 + 0.661T + 71T^{2} \) |
| 73 | \( 1 + 11.5T + 73T^{2} \) |
| 79 | \( 1 + 15.7T + 79T^{2} \) |
| 83 | \( 1 - 13.7T + 83T^{2} \) |
| 89 | \( 1 + 5.29T + 89T^{2} \) |
| 97 | \( 1 - 11.1T + 97T^{2} \) |
| 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
−7.31112703051233282617304194737, −6.36928915990224471048425012942, −5.92995343676606460899839364405, −5.54485309527627481991419042727, −4.91219698774426087049841924582, −4.11071744068249545480833742812, −3.17431455364828796688174420562, −2.04602781415431231535116688762, −1.50942449095690714030855860479, 0,
1.50942449095690714030855860479, 2.04602781415431231535116688762, 3.17431455364828796688174420562, 4.11071744068249545480833742812, 4.91219698774426087049841924582, 5.54485309527627481991419042727, 5.92995343676606460899839364405, 6.36928915990224471048425012942, 7.31112703051233282617304194737
