| L(s) = 1 | − 2.91·2-s + 3.10·3-s + 0.482·4-s + 2.69·5-s − 9.05·6-s − 7·7-s + 21.8·8-s − 17.3·9-s − 7.84·10-s − 6.61·11-s + 1.50·12-s + 21.3·13-s + 20.3·14-s + 8.37·15-s − 67.6·16-s − 117.·17-s + 50.4·18-s + 92.4·19-s + 1.29·20-s − 21.7·21-s + 19.2·22-s + 23·23-s + 68.0·24-s − 117.·25-s − 62.2·26-s − 137.·27-s − 3.37·28-s + ⋯ |
| L(s) = 1 | − 1.02·2-s + 0.598·3-s + 0.0603·4-s + 0.240·5-s − 0.616·6-s − 0.377·7-s + 0.967·8-s − 0.641·9-s − 0.248·10-s − 0.181·11-s + 0.0361·12-s + 0.455·13-s + 0.389·14-s + 0.144·15-s − 1.05·16-s − 1.67·17-s + 0.661·18-s + 1.11·19-s + 0.0145·20-s − 0.226·21-s + 0.186·22-s + 0.208·23-s + 0.578·24-s − 0.941·25-s − 0.469·26-s − 0.982·27-s − 0.0228·28-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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 2.91T + 8T^{2} \) |
| 3 | \( 1 - 3.10T + 27T^{2} \) |
| 5 | \( 1 - 2.69T + 125T^{2} \) |
| 11 | \( 1 + 6.61T + 1.33e3T^{2} \) |
| 13 | \( 1 - 21.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 117.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 92.4T + 6.85e3T^{2} \) |
| 29 | \( 1 + 137.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 245.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 78.4T + 5.06e4T^{2} \) |
| 41 | \( 1 + 226.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 557.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 278.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 114.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 311.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 449.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 881.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.04e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 83.8T + 3.89e5T^{2} \) |
| 79 | \( 1 + 622.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.41e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.24e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 727.T + 9.12e5T^{2} \) |
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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
−11.61069501311143101746733098383, −10.68495010866018817766846573744, −9.515827391495049308085830711473, −8.955469723167983299467799794152, −8.050866692663613009116519582053, −6.91531746629167002378486059094, −5.37316307535753319425271968292, −3.67009822240100017255205656157, −2.00072481162689884701107007771, 0,
2.00072481162689884701107007771, 3.67009822240100017255205656157, 5.37316307535753319425271968292, 6.91531746629167002378486059094, 8.050866692663613009116519582053, 8.955469723167983299467799794152, 9.515827391495049308085830711473, 10.68495010866018817766846573744, 11.61069501311143101746733098383
