| L(s) = 1 | − 4.85·2-s − 3·3-s + 15.5·4-s − 5·5-s + 14.5·6-s + 5.17·7-s − 36.5·8-s + 9·9-s + 24.2·10-s − 46.5·12-s − 43.6·13-s − 25.1·14-s + 15·15-s + 52.9·16-s + 96.8·17-s − 43.6·18-s − 40.3·19-s − 77.6·20-s − 15.5·21-s − 65.7·23-s + 109.·24-s + 25·25-s + 211.·26-s − 27·27-s + 80.4·28-s + 139.·29-s − 72.7·30-s + ⋯ |
| L(s) = 1 | − 1.71·2-s − 0.577·3-s + 1.94·4-s − 0.447·5-s + 0.990·6-s + 0.279·7-s − 1.61·8-s + 0.333·9-s + 0.766·10-s − 1.12·12-s − 0.930·13-s − 0.479·14-s + 0.258·15-s + 0.827·16-s + 1.38·17-s − 0.571·18-s − 0.487·19-s − 0.868·20-s − 0.161·21-s − 0.596·23-s + 0.932·24-s + 0.200·25-s + 1.59·26-s − 0.192·27-s + 0.542·28-s + 0.894·29-s − 0.442·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + 3T \) |
| 5 | \( 1 + 5T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 4.85T + 8T^{2} \) |
| 7 | \( 1 - 5.17T + 343T^{2} \) |
| 13 | \( 1 + 43.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 96.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 40.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 65.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 139.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 144.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 121.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 249.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 108.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 360.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 105.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 359.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 419.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 1.01e3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 628.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 74.2T + 3.89e5T^{2} \) |
| 79 | \( 1 - 663.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 60.4T + 5.71e5T^{2} \) |
| 89 | \( 1 - 284.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.85e3T + 9.12e5T^{2} \) |
| 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
−8.247427892688126718126297770361, −8.065913971648601071420616003371, −7.10744043614285073649698078785, −6.52630150663914970336782492709, −5.41869277799807338696361734291, −4.46689233697830723124724872842, −3.10690061002179931905737510972, −1.94441885797652149865794652644, −0.931448841170284947073451105518, 0,
0.931448841170284947073451105518, 1.94441885797652149865794652644, 3.10690061002179931905737510972, 4.46689233697830723124724872842, 5.41869277799807338696361734291, 6.52630150663914970336782492709, 7.10744043614285073649698078785, 8.065913971648601071420616003371, 8.247427892688126718126297770361
