| L(s) = 1 | − 9.72·2-s − 28.7·3-s + 62.5·4-s − 25·5-s + 279.·6-s − 28.8·7-s − 297.·8-s + 581.·9-s + 243.·10-s + 121·11-s − 1.79e3·12-s − 385.·13-s + 280.·14-s + 717.·15-s + 891.·16-s + 296.·17-s − 5.65e3·18-s − 361·19-s − 1.56e3·20-s + 827.·21-s − 1.17e3·22-s − 3.76e3·23-s + 8.54e3·24-s + 625·25-s + 3.74e3·26-s − 9.71e3·27-s − 1.80e3·28-s + ⋯ |
| L(s) = 1 | − 1.71·2-s − 1.84·3-s + 1.95·4-s − 0.447·5-s + 3.16·6-s − 0.222·7-s − 1.64·8-s + 2.39·9-s + 0.768·10-s + 0.301·11-s − 3.60·12-s − 0.632·13-s + 0.382·14-s + 0.823·15-s + 0.870·16-s + 0.248·17-s − 4.11·18-s − 0.229·19-s − 0.874·20-s + 0.409·21-s − 0.518·22-s − 1.48·23-s + 3.02·24-s + 0.200·25-s + 1.08·26-s − 2.56·27-s − 0.434·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.3133555548\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3133555548\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 + 361T \) |
| good | 2 | \( 1 + 9.72T + 32T^{2} \) |
| 3 | \( 1 + 28.7T + 243T^{2} \) |
| 7 | \( 1 + 28.8T + 1.68e4T^{2} \) |
| 13 | \( 1 + 385.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 296.T + 1.41e6T^{2} \) |
| 23 | \( 1 + 3.76e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.92e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.45e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.11e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 2.31e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.34e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.16e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.53e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.04e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.33e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.13e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.86e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.47e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.02e5T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.17e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.09e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 9.78e4T + 8.58e9T^{2} \) |
| 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
−9.507768572055232819307070572914, −8.253450951412613812216813344699, −7.60222531631717772231879012092, −6.62487306747379576989074008372, −6.28944736708046120124471624647, −5.08197610365832970719204393231, −4.07510094274172563416971230584, −2.31239028396566318523147783338, −1.06213686321669940912587628347, −0.44049808586085997616757652705,
0.44049808586085997616757652705, 1.06213686321669940912587628347, 2.31239028396566318523147783338, 4.07510094274172563416971230584, 5.08197610365832970719204393231, 6.28944736708046120124471624647, 6.62487306747379576989074008372, 7.60222531631717772231879012092, 8.253450951412613812216813344699, 9.507768572055232819307070572914
