| L(s) = 1 | − 0.428·3-s − 3.38·5-s + 7-s − 2.81·9-s + 0.627·11-s + 2.39·13-s + 1.45·15-s − 0.176·17-s − 3.66·19-s − 0.428·21-s − 0.00146·23-s + 6.42·25-s + 2.49·27-s − 8.21·29-s + 9.63·31-s − 0.269·33-s − 3.38·35-s − 0.935·37-s − 1.02·39-s + 1.52·41-s + 11.5·43-s + 9.51·45-s + 6.00·47-s + 49-s + 0.0756·51-s + 6.37·53-s − 2.12·55-s + ⋯ |
| L(s) = 1 | − 0.247·3-s − 1.51·5-s + 0.377·7-s − 0.938·9-s + 0.189·11-s + 0.664·13-s + 0.374·15-s − 0.0427·17-s − 0.839·19-s − 0.0936·21-s − 0.000305·23-s + 1.28·25-s + 0.480·27-s − 1.52·29-s + 1.73·31-s − 0.0468·33-s − 0.571·35-s − 0.153·37-s − 0.164·39-s + 0.238·41-s + 1.75·43-s + 1.41·45-s + 0.875·47-s + 0.142·49-s + 0.0105·51-s + 0.875·53-s − 0.286·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 + 0.428T + 3T^{2} \) |
| 5 | \( 1 + 3.38T + 5T^{2} \) |
| 11 | \( 1 - 0.627T + 11T^{2} \) |
| 13 | \( 1 - 2.39T + 13T^{2} \) |
| 17 | \( 1 + 0.176T + 17T^{2} \) |
| 19 | \( 1 + 3.66T + 19T^{2} \) |
| 23 | \( 1 + 0.00146T + 23T^{2} \) |
| 29 | \( 1 + 8.21T + 29T^{2} \) |
| 31 | \( 1 - 9.63T + 31T^{2} \) |
| 37 | \( 1 + 0.935T + 37T^{2} \) |
| 41 | \( 1 - 1.52T + 41T^{2} \) |
| 43 | \( 1 - 11.5T + 43T^{2} \) |
| 47 | \( 1 - 6.00T + 47T^{2} \) |
| 53 | \( 1 - 6.37T + 53T^{2} \) |
| 59 | \( 1 - 0.395T + 59T^{2} \) |
| 61 | \( 1 + 2.25T + 61T^{2} \) |
| 67 | \( 1 + 5.83T + 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 + 0.479T + 73T^{2} \) |
| 79 | \( 1 - 9.12T + 79T^{2} \) |
| 83 | \( 1 + 11.9T + 83T^{2} \) |
| 89 | \( 1 + 5.88T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−7.63395387079880254544369394246, −7.01550101030697034501546519925, −6.09283206300833737722140139238, −5.57398528732381524441474665723, −4.46138988012006108392992662942, −4.11020068351280272278304505329, −3.25539431050583008397001832690, −2.38867316336787034498223826008, −1.02221093824345394194953796457, 0,
1.02221093824345394194953796457, 2.38867316336787034498223826008, 3.25539431050583008397001832690, 4.11020068351280272278304505329, 4.46138988012006108392992662942, 5.57398528732381524441474665723, 6.09283206300833737722140139238, 7.01550101030697034501546519925, 7.63395387079880254544369394246
