| L(s) = 1 | − 3-s + 2.07·5-s − 1.37·7-s + 9-s + 5.27·11-s − 6.01·13-s − 2.07·15-s + 0.202·17-s + 0.336·19-s + 1.37·21-s + 2.04·23-s − 0.706·25-s − 27-s − 6.05·29-s + 5.31·31-s − 5.27·33-s − 2.84·35-s + 1.31·37-s + 6.01·39-s − 11.1·41-s + 0.965·43-s + 2.07·45-s − 4.19·47-s − 5.11·49-s − 0.202·51-s + 8.35·53-s + 10.9·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.926·5-s − 0.519·7-s + 0.333·9-s + 1.59·11-s − 1.66·13-s − 0.535·15-s + 0.0491·17-s + 0.0772·19-s + 0.299·21-s + 0.426·23-s − 0.141·25-s − 0.192·27-s − 1.12·29-s + 0.955·31-s − 0.918·33-s − 0.481·35-s + 0.216·37-s + 0.962·39-s − 1.74·41-s + 0.147·43-s + 0.308·45-s − 0.612·47-s − 0.730·49-s − 0.0283·51-s + 1.14·53-s + 1.47·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 - 2.07T + 5T^{2} \) |
| 7 | \( 1 + 1.37T + 7T^{2} \) |
| 11 | \( 1 - 5.27T + 11T^{2} \) |
| 13 | \( 1 + 6.01T + 13T^{2} \) |
| 17 | \( 1 - 0.202T + 17T^{2} \) |
| 19 | \( 1 - 0.336T + 19T^{2} \) |
| 23 | \( 1 - 2.04T + 23T^{2} \) |
| 29 | \( 1 + 6.05T + 29T^{2} \) |
| 31 | \( 1 - 5.31T + 31T^{2} \) |
| 37 | \( 1 - 1.31T + 37T^{2} \) |
| 41 | \( 1 + 11.1T + 41T^{2} \) |
| 43 | \( 1 - 0.965T + 43T^{2} \) |
| 47 | \( 1 + 4.19T + 47T^{2} \) |
| 53 | \( 1 - 8.35T + 53T^{2} \) |
| 59 | \( 1 + 0.730T + 59T^{2} \) |
| 61 | \( 1 + 10.2T + 61T^{2} \) |
| 67 | \( 1 + 0.951T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 - 15.3T + 79T^{2} \) |
| 83 | \( 1 + 0.663T + 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 - 2.07T + 97T^{2} \) |
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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
−7.20436851728650111601977769073, −6.72032461897773524108066927955, −6.18417760015034488160835280873, −5.43276679339024348363790462101, −4.80268555701265040425928261624, −3.98190898520338456372770195152, −3.07125174144606991826315152230, −2.09817366027946714201123983566, −1.31363043126904502381988635252, 0,
1.31363043126904502381988635252, 2.09817366027946714201123983566, 3.07125174144606991826315152230, 3.98190898520338456372770195152, 4.80268555701265040425928261624, 5.43276679339024348363790462101, 6.18417760015034488160835280873, 6.72032461897773524108066927955, 7.20436851728650111601977769073
