L(s) = 1 | − 8.96·2-s + 9·3-s + 48.3·4-s − 80.6·6-s − 191.·7-s − 146.·8-s + 81·9-s − 121·11-s + 434.·12-s + 257.·13-s + 1.71e3·14-s − 235.·16-s + 1.24e3·17-s − 725.·18-s − 2.95e3·19-s − 1.72e3·21-s + 1.08e3·22-s − 374.·23-s − 1.31e3·24-s − 2.30e3·26-s + 729·27-s − 9.25e3·28-s + 4.57e3·29-s + 2.52e3·31-s + 6.79e3·32-s − 1.08e3·33-s − 1.11e4·34-s + ⋯ |
L(s) = 1 | − 1.58·2-s + 0.577·3-s + 1.50·4-s − 0.914·6-s − 1.47·7-s − 0.807·8-s + 0.333·9-s − 0.301·11-s + 0.871·12-s + 0.422·13-s + 2.33·14-s − 0.230·16-s + 1.04·17-s − 0.528·18-s − 1.87·19-s − 0.852·21-s + 0.477·22-s − 0.147·23-s − 0.466·24-s − 0.669·26-s + 0.192·27-s − 2.22·28-s + 1.00·29-s + 0.471·31-s + 1.17·32-s − 0.174·33-s − 1.65·34-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 | \( 1 + 8.96T + 32T^{2} \) |
| 7 | \( 1 + 191.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 257.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.24e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.95e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 374.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.57e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.52e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.13e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 7.60e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 853.T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.85e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.55e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.39e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.13e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.53e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.20e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.72e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.51e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.02e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.13e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 2.33e4T + 8.58e9T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.978994604716769751846751843396, −8.428479505026643134800964182679, −7.62632147991448470029201392787, −6.67679600068368515640504446458, −6.06386313576387877039205331174, −4.31622949854532930839124692631, −3.13336300002015501473016062560, −2.27769626753586617621426201637, −0.994401446011749395080638946479, 0,
0.994401446011749395080638946479, 2.27769626753586617621426201637, 3.13336300002015501473016062560, 4.31622949854532930839124692631, 6.06386313576387877039205331174, 6.67679600068368515640504446458, 7.62632147991448470029201392787, 8.428479505026643134800964182679, 8.978994604716769751846751843396