| L(s) = 1 | − 9.37·2-s − 9·3-s + 55.8·4-s + 84.3·6-s + 105.·7-s − 223.·8-s + 81·9-s − 121·11-s − 502.·12-s − 147.·13-s − 984.·14-s + 307.·16-s + 1.43e3·17-s − 759.·18-s + 2.03e3·19-s − 945.·21-s + 1.13e3·22-s − 828.·23-s + 2.01e3·24-s + 1.38e3·26-s − 729·27-s + 5.86e3·28-s + 4.63e3·29-s − 9.83e3·31-s + 4.27e3·32-s + 1.08e3·33-s − 1.34e4·34-s + ⋯ |
| L(s) = 1 | − 1.65·2-s − 0.577·3-s + 1.74·4-s + 0.956·6-s + 0.810·7-s − 1.23·8-s + 0.333·9-s − 0.301·11-s − 1.00·12-s − 0.242·13-s − 1.34·14-s + 0.299·16-s + 1.20·17-s − 0.552·18-s + 1.29·19-s − 0.467·21-s + 0.499·22-s − 0.326·23-s + 0.712·24-s + 0.401·26-s − 0.192·27-s + 1.41·28-s + 1.02·29-s − 1.83·31-s + 0.737·32-s + 0.174·33-s − 1.99·34-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 | \( 1 + 9.37T + 32T^{2} \) |
| 7 | \( 1 - 105.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 147.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.43e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.03e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 828.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.63e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 9.83e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 7.13e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.82e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.38e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.29e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.43e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 7.08e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.84e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.62e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.81e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.93e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.12e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.33e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.03e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.49e5T + 8.58e9T^{2} \) |
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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
−9.183083562493198373111822261516, −8.057035185994303022535504120838, −7.68218227638774868274934247614, −6.81364505083086098657355616177, −5.64926142722997867984714934937, −4.81725415842755290482740904813, −3.22607153611287515436809177572, −1.82496914453194142174581907831, −1.08269092503715194022116730788, 0,
1.08269092503715194022116730788, 1.82496914453194142174581907831, 3.22607153611287515436809177572, 4.81725415842755290482740904813, 5.64926142722997867984714934937, 6.81364505083086098657355616177, 7.68218227638774868274934247614, 8.057035185994303022535504120838, 9.183083562493198373111822261516
