L(s) = 1 | − 14.2·2-s − 27·3-s + 75.7·4-s + 109.·5-s + 385.·6-s + 411.·7-s + 746.·8-s + 729·9-s − 1.56e3·10-s + 1.33e3·11-s − 2.04e3·12-s − 7.18e3·13-s − 5.87e3·14-s − 2.96e3·15-s − 2.03e4·16-s − 7.78e3·17-s − 1.04e4·18-s − 2.79e4·19-s + 8.31e3·20-s − 1.11e4·21-s − 1.89e4·22-s + 2.34e4·23-s − 2.01e4·24-s − 6.60e4·25-s + 1.02e5·26-s − 1.96e4·27-s + 3.11e4·28-s + ⋯ |
L(s) = 1 | − 1.26·2-s − 0.577·3-s + 0.591·4-s + 0.392·5-s + 0.728·6-s + 0.453·7-s + 0.515·8-s + 0.333·9-s − 0.495·10-s + 0.301·11-s − 0.341·12-s − 0.906·13-s − 0.572·14-s − 0.226·15-s − 1.24·16-s − 0.384·17-s − 0.420·18-s − 0.934·19-s + 0.232·20-s − 0.261·21-s − 0.380·22-s + 0.401·23-s − 0.297·24-s − 0.845·25-s + 1.14·26-s − 0.192·27-s + 0.268·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 27T \) |
| 11 | \( 1 - 1.33e3T \) |
good | 2 | \( 1 + 14.2T + 128T^{2} \) |
| 5 | \( 1 - 109.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 411.T + 8.23e5T^{2} \) |
| 13 | \( 1 + 7.18e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 7.78e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.79e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 2.34e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 7.50e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.67e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 5.65e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + 6.10e4T + 1.94e11T^{2} \) |
| 43 | \( 1 + 7.45e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.22e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 6.82e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 6.64e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.01e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.66e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.80e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 9.68e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.25e5T + 1.92e13T^{2} \) |
| 83 | \( 1 - 8.19e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 4.48e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.20e7T + 8.07e13T^{2} \) |
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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
−14.76083765316836704468180056454, −13.25479144991627693396392748898, −11.68867612192354316618897898708, −10.50273010917865603031675098429, −9.480695254696126452903927650318, −8.132485007663403948096680305629, −6.69127306155393940452628761948, −4.78335484458172153954764865401, −1.75715288048953095286512485367, 0,
1.75715288048953095286512485367, 4.78335484458172153954764865401, 6.69127306155393940452628761948, 8.132485007663403948096680305629, 9.480695254696126452903927650318, 10.50273010917865603031675098429, 11.68867612192354316618897898708, 13.25479144991627693396392748898, 14.76083765316836704468180056454