| L(s) = 1 | + 0.875·2-s − 3·3-s − 7.23·4-s − 5·5-s − 2.62·6-s + 27.5·7-s − 13.3·8-s + 9·9-s − 4.37·10-s + 21.6·12-s − 55.3·13-s + 24.1·14-s + 15·15-s + 46.1·16-s − 61.3·17-s + 7.88·18-s + 79.3·19-s + 36.1·20-s − 82.6·21-s − 16.3·23-s + 40.0·24-s + 25·25-s − 48.4·26-s − 27·27-s − 199.·28-s + 32.8·29-s + 13.1·30-s + ⋯ |
| L(s) = 1 | + 0.309·2-s − 0.577·3-s − 0.904·4-s − 0.447·5-s − 0.178·6-s + 1.48·7-s − 0.589·8-s + 0.333·9-s − 0.138·10-s + 0.522·12-s − 1.18·13-s + 0.460·14-s + 0.258·15-s + 0.721·16-s − 0.874·17-s + 0.103·18-s + 0.957·19-s + 0.404·20-s − 0.858·21-s − 0.147·23-s + 0.340·24-s + 0.200·25-s − 0.365·26-s − 0.192·27-s − 1.34·28-s + 0.210·29-s + 0.0799·30-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 | \( 1 - 0.875T + 8T^{2} \) |
| 7 | \( 1 - 27.5T + 343T^{2} \) |
| 13 | \( 1 + 55.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 61.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 79.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 16.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 32.8T + 2.43e4T^{2} \) |
| 31 | \( 1 + 7.60T + 2.97e4T^{2} \) |
| 37 | \( 1 + 410.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 317.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 345.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 21.0T + 1.03e5T^{2} \) |
| 53 | \( 1 + 156.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 773.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 838.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 514.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 657.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 779.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 357.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.27e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.35e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 812.T + 9.12e5T^{2} \) |
| 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
−8.514285954609732216048145933780, −7.68493525564760270853401273582, −7.08150497480256467383971394251, −5.77082603640432011149374202529, −5.03640709259825310594406374581, −4.61433790885154065695409931426, −3.77709961627283169376615764080, −2.39185479333433798345106971170, −1.09237787237170818919001466511, 0,
1.09237787237170818919001466511, 2.39185479333433798345106971170, 3.77709961627283169376615764080, 4.61433790885154065695409931426, 5.03640709259825310594406374581, 5.77082603640432011149374202529, 7.08150497480256467383971394251, 7.68493525564760270853401273582, 8.514285954609732216048145933780
