| L(s) = 1 | + 3-s + 5-s − 3·7-s + 9-s − 6·11-s + 2·13-s + 15-s + 3·17-s − 6·19-s − 3·21-s + 23-s + 25-s + 27-s + 9·29-s + 3·31-s − 6·33-s − 3·35-s − 3·37-s + 2·39-s − 3·41-s + 45-s − 4·47-s + 2·49-s + 3·51-s + 9·53-s − 6·55-s − 6·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.13·7-s + 1/3·9-s − 1.80·11-s + 0.554·13-s + 0.258·15-s + 0.727·17-s − 1.37·19-s − 0.654·21-s + 0.208·23-s + 1/5·25-s + 0.192·27-s + 1.67·29-s + 0.538·31-s − 1.04·33-s − 0.507·35-s − 0.493·37-s + 0.320·39-s − 0.468·41-s + 0.149·45-s − 0.583·47-s + 2/7·49-s + 0.420·51-s + 1.23·53-s − 0.809·55-s − 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−15.81724842179749, −15.28961692490550, −14.83690517497566, −13.95999472509248, −13.63866188999215, −13.13801248821409, −12.61304102811906, −12.35256069137809, −11.33871541327184, −10.58718625928042, −10.18942073439572, −9.943490318519978, −9.123559766903533, −8.419638977556713, −8.186345281436861, −7.402682458180598, −6.512032340642790, −6.398622338595669, −5.387511132818302, −4.970002912689382, −4.008437345471037, −3.326455871651505, −2.672448922622752, −2.257569267464834, −1.061881946334311, 0,
1.061881946334311, 2.257569267464834, 2.672448922622752, 3.326455871651505, 4.008437345471037, 4.970002912689382, 5.387511132818302, 6.398622338595669, 6.512032340642790, 7.402682458180598, 8.186345281436861, 8.419638977556713, 9.123559766903533, 9.943490318519978, 10.18942073439572, 10.58718625928042, 11.33871541327184, 12.35256069137809, 12.61304102811906, 13.13801248821409, 13.63866188999215, 13.95999472509248, 14.83690517497566, 15.28961692490550, 15.81724842179749
