| L(s) = 1 | + 2.11·2-s − 9.53·3-s − 3.51·4-s − 13.8·5-s − 20.2·6-s + 7·7-s − 24.3·8-s + 64.0·9-s − 29.3·10-s + 44.5·11-s + 33.4·12-s − 14.0·13-s + 14.8·14-s + 132.·15-s − 23.5·16-s + 73.5·17-s + 135.·18-s − 144.·19-s + 48.6·20-s − 66.7·21-s + 94.2·22-s + 23·23-s + 232.·24-s + 66.8·25-s − 29.7·26-s − 352.·27-s − 24.5·28-s + ⋯ |
| L(s) = 1 | + 0.749·2-s − 1.83·3-s − 0.438·4-s − 1.23·5-s − 1.37·6-s + 0.377·7-s − 1.07·8-s + 2.37·9-s − 0.928·10-s + 1.21·11-s + 0.805·12-s − 0.299·13-s + 0.283·14-s + 2.27·15-s − 0.368·16-s + 1.04·17-s + 1.77·18-s − 1.74·19-s + 0.543·20-s − 0.693·21-s + 0.913·22-s + 0.208·23-s + 1.97·24-s + 0.535·25-s − 0.224·26-s − 2.51·27-s − 0.165·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7583306352\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7583306352\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 7T \) |
| 23 | \( 1 - 23T \) |
| good | 2 | \( 1 - 2.11T + 8T^{2} \) |
| 3 | \( 1 + 9.53T + 27T^{2} \) |
| 5 | \( 1 + 13.8T + 125T^{2} \) |
| 11 | \( 1 - 44.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 14.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 73.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 144.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 151.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 267.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 217.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 292.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 258.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 134.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 358.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 726.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 22.6T + 2.26e5T^{2} \) |
| 67 | \( 1 - 441.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 42.7T + 3.57e5T^{2} \) |
| 73 | \( 1 + 414.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.24e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 235.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 229.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 548.T + 9.12e5T^{2} \) |
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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
−12.23137791996109344869070449771, −11.68991067162480322662783153240, −10.90613259181043581359288881480, −9.570298350380192078944310701622, −8.047375685008354031018159752901, −6.69696682142711672700099264381, −5.76082497388056108456869102233, −4.52968510417035055692680212116, −4.00312890749249220447446320083, −0.69125718611234209828944770447,
0.69125718611234209828944770447, 4.00312890749249220447446320083, 4.52968510417035055692680212116, 5.76082497388056108456869102233, 6.69696682142711672700099264381, 8.047375685008354031018159752901, 9.570298350380192078944310701622, 10.90613259181043581359288881480, 11.68991067162480322662783153240, 12.23137791996109344869070449771
