| L(s) = 1 | + 3-s + 3.86·5-s − 2.44·7-s + 9-s − 1.46·11-s + 4.24·13-s + 3.86·15-s − 3.46·17-s + 7.46·19-s − 2.44·21-s − 2.82·23-s + 9.92·25-s + 27-s + 8.76·29-s − 7.34·31-s − 1.46·33-s − 9.46·35-s + 0.656·37-s + 4.24·39-s + 4.53·41-s − 3.46·43-s + 3.86·45-s + 2.82·47-s − 1.00·49-s − 3.46·51-s − 4.62·53-s − 5.65·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.72·5-s − 0.925·7-s + 0.333·9-s − 0.441·11-s + 1.17·13-s + 0.997·15-s − 0.840·17-s + 1.71·19-s − 0.534·21-s − 0.589·23-s + 1.98·25-s + 0.192·27-s + 1.62·29-s − 1.31·31-s − 0.254·33-s − 1.59·35-s + 0.107·37-s + 0.679·39-s + 0.708·41-s − 0.528·43-s + 0.575·45-s + 0.412·47-s − 0.142·49-s − 0.485·51-s − 0.634·53-s − 0.762·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.111753071\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.111753071\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| good | 5 | \( 1 - 3.86T + 5T^{2} \) |
| 7 | \( 1 + 2.44T + 7T^{2} \) |
| 11 | \( 1 + 1.46T + 11T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 - 7.46T + 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 - 8.76T + 29T^{2} \) |
| 31 | \( 1 + 7.34T + 31T^{2} \) |
| 37 | \( 1 - 0.656T + 37T^{2} \) |
| 41 | \( 1 - 4.53T + 41T^{2} \) |
| 43 | \( 1 + 3.46T + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 + 4.62T + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 + 0.656T + 61T^{2} \) |
| 67 | \( 1 - 14.9T + 67T^{2} \) |
| 71 | \( 1 + 6.41T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 - 2.44T + 79T^{2} \) |
| 83 | \( 1 - 5.46T + 83T^{2} \) |
| 89 | \( 1 + 4.92T + 89T^{2} \) |
| 97 | \( 1 - 1.07T + 97T^{2} \) |
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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
−8.885046413659957192611514967172, −8.122898828661727524025422314989, −7.01473946104793750466200384402, −6.42207283129074213461975864870, −5.74899536970872883666464557580, −5.03858634664365566238766702091, −3.76554254503579320989267947350, −2.93941496353539956998599925223, −2.17643494284857105043127354728, −1.11270505145013229578390795304,
1.11270505145013229578390795304, 2.17643494284857105043127354728, 2.93941496353539956998599925223, 3.76554254503579320989267947350, 5.03858634664365566238766702091, 5.74899536970872883666464557580, 6.42207283129074213461975864870, 7.01473946104793750466200384402, 8.122898828661727524025422314989, 8.885046413659957192611514967172
