L(s) = 1 | + 0.870·2-s − 9·3-s − 31.2·4-s − 7.83·6-s + 236.·7-s − 55.0·8-s + 81·9-s − 121·11-s + 281.·12-s + 261.·13-s + 205.·14-s + 951.·16-s − 1.58e3·17-s + 70.5·18-s − 711.·19-s − 2.12e3·21-s − 105.·22-s − 2.96e3·23-s + 495.·24-s + 227.·26-s − 729·27-s − 7.39e3·28-s + 4.42e3·29-s − 4.44e3·31-s + 2.59e3·32-s + 1.08e3·33-s − 1.38e3·34-s + ⋯ |
L(s) = 1 | + 0.153·2-s − 0.577·3-s − 0.976·4-s − 0.0888·6-s + 1.82·7-s − 0.304·8-s + 0.333·9-s − 0.301·11-s + 0.563·12-s + 0.429·13-s + 0.280·14-s + 0.929·16-s − 1.33·17-s + 0.0512·18-s − 0.452·19-s − 1.05·21-s − 0.0463·22-s − 1.16·23-s + 0.175·24-s + 0.0661·26-s − 0.192·27-s − 1.78·28-s + 0.976·29-s − 0.830·31-s + 0.447·32-s + 0.174·33-s − 0.204·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 9T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 121T \) |
good | 2 | \( 1 - 0.870T + 32T^{2} \) |
| 7 | \( 1 - 236.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 261.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.58e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 711.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.96e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.42e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.44e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.16e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 4.68e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.01e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 8.28e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.44e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.44e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.72e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.15e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.67e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.90e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.15e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.07e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.84e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 5.03e4T + 8.58e9T^{2} \) |
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
−8.800246693540748713388777378019, −8.358198067060178760688752975487, −7.48539753110778961871197749160, −6.20417049558566020543114449205, −5.33450337407057342253206061062, −4.55965843889216518274870451233, −4.03902410577290157575518014833, −2.24992140439565892788638824351, −1.16289169987838982676389286958, 0,
1.16289169987838982676389286958, 2.24992140439565892788638824351, 4.03902410577290157575518014833, 4.55965843889216518274870451233, 5.33450337407057342253206061062, 6.20417049558566020543114449205, 7.48539753110778961871197749160, 8.358198067060178760688752975487, 8.800246693540748713388777378019