| L(s) = 1 | + 0.0957·3-s + 3.87·5-s − 3.14·7-s − 2.99·9-s − 1.99·11-s + 3.58·13-s + 0.370·15-s − 2.39·17-s − 4.12·19-s − 0.300·21-s + 2.01·23-s + 10.0·25-s − 0.573·27-s + 8.01·29-s − 6.88·31-s − 0.190·33-s − 12.1·35-s − 2.67·37-s + 0.343·39-s − 2.97·41-s + 5.35·43-s − 11.5·45-s + 11.1·47-s + 2.88·49-s − 0.229·51-s + 4.98·53-s − 7.71·55-s + ⋯ |
| L(s) = 1 | + 0.0552·3-s + 1.73·5-s − 1.18·7-s − 0.996·9-s − 0.600·11-s + 0.993·13-s + 0.0957·15-s − 0.580·17-s − 0.945·19-s − 0.0656·21-s + 0.421·23-s + 2.00·25-s − 0.110·27-s + 1.48·29-s − 1.23·31-s − 0.0331·33-s − 2.05·35-s − 0.438·37-s + 0.0549·39-s − 0.465·41-s + 0.817·43-s − 1.72·45-s + 1.62·47-s + 0.411·49-s − 0.0320·51-s + 0.684·53-s − 1.03·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 251 | \( 1 + T \) |
| good | 3 | \( 1 - 0.0957T + 3T^{2} \) |
| 5 | \( 1 - 3.87T + 5T^{2} \) |
| 7 | \( 1 + 3.14T + 7T^{2} \) |
| 11 | \( 1 + 1.99T + 11T^{2} \) |
| 13 | \( 1 - 3.58T + 13T^{2} \) |
| 17 | \( 1 + 2.39T + 17T^{2} \) |
| 19 | \( 1 + 4.12T + 19T^{2} \) |
| 23 | \( 1 - 2.01T + 23T^{2} \) |
| 29 | \( 1 - 8.01T + 29T^{2} \) |
| 31 | \( 1 + 6.88T + 31T^{2} \) |
| 37 | \( 1 + 2.67T + 37T^{2} \) |
| 41 | \( 1 + 2.97T + 41T^{2} \) |
| 43 | \( 1 - 5.35T + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 - 4.98T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 + 6.37T + 61T^{2} \) |
| 67 | \( 1 + 11.1T + 67T^{2} \) |
| 71 | \( 1 - 6.00T + 71T^{2} \) |
| 73 | \( 1 + 3.84T + 73T^{2} \) |
| 79 | \( 1 - 6.21T + 79T^{2} \) |
| 83 | \( 1 + 6.53T + 83T^{2} \) |
| 89 | \( 1 + 12.5T + 89T^{2} \) |
| 97 | \( 1 + 5.14T + 97T^{2} \) |
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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
−7.31872814870295486895575159396, −6.48588451117761378452576532986, −6.12185427017670452915255156994, −5.65235973959240404112941645163, −4.85548970949497581466406694735, −3.75890294275021327221688424538, −2.80334289581955461427249376131, −2.46491874634110723148535205776, −1.36406081245587558830248277839, 0,
1.36406081245587558830248277839, 2.46491874634110723148535205776, 2.80334289581955461427249376131, 3.75890294275021327221688424538, 4.85548970949497581466406694735, 5.65235973959240404112941645163, 6.12185427017670452915255156994, 6.48588451117761378452576532986, 7.31872814870295486895575159396
