| L(s) = 1 | − 1.44·2-s + 9·3-s − 29.9·4-s − 25·5-s − 13.0·6-s + 41.5·7-s + 89.5·8-s + 81·9-s + 36.1·10-s + 121·11-s − 269.·12-s + 434.·13-s − 60.0·14-s − 225·15-s + 827.·16-s + 474.·17-s − 117.·18-s − 2.58e3·19-s + 747.·20-s + 373.·21-s − 175.·22-s − 3.67e3·23-s + 806.·24-s + 625·25-s − 628.·26-s + 729·27-s − 1.24e3·28-s + ⋯ |
| L(s) = 1 | − 0.255·2-s + 0.577·3-s − 0.934·4-s − 0.447·5-s − 0.147·6-s + 0.320·7-s + 0.494·8-s + 0.333·9-s + 0.114·10-s + 0.301·11-s − 0.539·12-s + 0.713·13-s − 0.0819·14-s − 0.258·15-s + 0.807·16-s + 0.398·17-s − 0.0852·18-s − 1.64·19-s + 0.417·20-s + 0.184·21-s − 0.0771·22-s − 1.44·23-s + 0.285·24-s + 0.200·25-s − 0.182·26-s + 0.192·27-s − 0.299·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 25T \) |
| 11 | \( 1 - 121T \) |
| good | 2 | \( 1 + 1.44T + 32T^{2} \) |
| 7 | \( 1 - 41.5T + 1.68e4T^{2} \) |
| 13 | \( 1 - 434.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 474.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.58e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.67e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.26e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 8.41e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.39e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.77e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.14e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.10e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.09e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.30e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.23e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.88e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.04e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.33e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 9.53e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.66e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.43e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.28e5T + 8.58e9T^{2} \) |
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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
−11.33639725029993699298873867625, −10.26432736657659890824356406804, −9.196263354115739681207289537677, −8.382083376923113653343472722928, −7.63361200949124834980424528031, −6.01252342373872770637520303925, −4.45326433144168677446687328834, −3.64463929107714971973204415576, −1.68659459446184327513572108682, 0,
1.68659459446184327513572108682, 3.64463929107714971973204415576, 4.45326433144168677446687328834, 6.01252342373872770637520303925, 7.63361200949124834980424528031, 8.382083376923113653343472722928, 9.196263354115739681207289537677, 10.26432736657659890824356406804, 11.33639725029993699298873867625
