| L(s) = 1 | + 1.40·2-s + 8.50·3-s − 6.01·4-s + 10.6·5-s + 11.9·6-s − 26.4·7-s − 19.7·8-s + 45.3·9-s + 14.9·10-s − 11·11-s − 51.1·12-s − 37.2·14-s + 90.2·15-s + 20.2·16-s + 3.00·17-s + 63.8·18-s + 164.·19-s − 63.8·20-s − 224.·21-s − 15.5·22-s − 194.·23-s − 167.·24-s − 12.3·25-s + 155.·27-s + 158.·28-s − 11.1·29-s + 127.·30-s + ⋯ |
| L(s) = 1 | + 0.498·2-s + 1.63·3-s − 0.751·4-s + 0.949·5-s + 0.815·6-s − 1.42·7-s − 0.872·8-s + 1.67·9-s + 0.473·10-s − 0.301·11-s − 1.23·12-s − 0.710·14-s + 1.55·15-s + 0.316·16-s + 0.0428·17-s + 0.836·18-s + 1.98·19-s − 0.713·20-s − 2.33·21-s − 0.150·22-s − 1.76·23-s − 1.42·24-s − 0.0991·25-s + 1.11·27-s + 1.07·28-s − 0.0713·29-s + 0.774·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.40T + 8T^{2} \) |
| 3 | \( 1 - 8.50T + 27T^{2} \) |
| 5 | \( 1 - 10.6T + 125T^{2} \) |
| 7 | \( 1 + 26.4T + 343T^{2} \) |
| 17 | \( 1 - 3.00T + 4.91e3T^{2} \) |
| 19 | \( 1 - 164.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 194.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 11.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 326.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 179.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 501.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 334.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 298.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 284.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 525.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 177.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 365.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 225.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 76.3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 886.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 247.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.18e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 420.T + 9.12e5T^{2} \) |
| 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
−8.651002046684958410845067827117, −7.83992969783945076593264881538, −6.96630196713096935950183403311, −5.90416201092009023700813766629, −5.31181456000849014019682860456, −3.93950505163050128253088690986, −3.41221842561522648766497640264, −2.73069589731527154703998628464, −1.65732842884487028907108437018, 0,
1.65732842884487028907108437018, 2.73069589731527154703998628464, 3.41221842561522648766497640264, 3.93950505163050128253088690986, 5.31181456000849014019682860456, 5.90416201092009023700813766629, 6.96630196713096935950183403311, 7.83992969783945076593264881538, 8.651002046684958410845067827117
