| L(s) = 1 | − 0.151·3-s + 3.67·5-s + 0.901·7-s − 2.97·9-s − 1.75·11-s + 13-s − 0.556·15-s − 5.39·17-s − 2.14·19-s − 0.136·21-s − 4.92·23-s + 8.52·25-s + 0.905·27-s + 29-s − 3.97·31-s + 0.265·33-s + 3.31·35-s + 2.48·37-s − 0.151·39-s + 0.995·41-s + 2.34·43-s − 10.9·45-s − 6.61·47-s − 6.18·49-s + 0.817·51-s + 14.0·53-s − 6.45·55-s + ⋯ |
| L(s) = 1 | − 0.0874·3-s + 1.64·5-s + 0.340·7-s − 0.992·9-s − 0.529·11-s + 0.277·13-s − 0.143·15-s − 1.30·17-s − 0.493·19-s − 0.0297·21-s − 1.02·23-s + 1.70·25-s + 0.174·27-s + 0.185·29-s − 0.713·31-s + 0.0462·33-s + 0.560·35-s + 0.407·37-s − 0.0242·39-s + 0.155·41-s + 0.357·43-s − 1.63·45-s − 0.965·47-s − 0.884·49-s + 0.114·51-s + 1.93·53-s − 0.870·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 0.151T + 3T^{2} \) |
| 5 | \( 1 - 3.67T + 5T^{2} \) |
| 7 | \( 1 - 0.901T + 7T^{2} \) |
| 11 | \( 1 + 1.75T + 11T^{2} \) |
| 17 | \( 1 + 5.39T + 17T^{2} \) |
| 19 | \( 1 + 2.14T + 19T^{2} \) |
| 23 | \( 1 + 4.92T + 23T^{2} \) |
| 31 | \( 1 + 3.97T + 31T^{2} \) |
| 37 | \( 1 - 2.48T + 37T^{2} \) |
| 41 | \( 1 - 0.995T + 41T^{2} \) |
| 43 | \( 1 - 2.34T + 43T^{2} \) |
| 47 | \( 1 + 6.61T + 47T^{2} \) |
| 53 | \( 1 - 14.0T + 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 + 1.78T + 61T^{2} \) |
| 67 | \( 1 + 14.3T + 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 - 7.83T + 73T^{2} \) |
| 79 | \( 1 + 6.15T + 79T^{2} \) |
| 83 | \( 1 - 14.3T + 83T^{2} \) |
| 89 | \( 1 + 0.813T + 89T^{2} \) |
| 97 | \( 1 + 15.5T + 97T^{2} \) |
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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
−7.83541202961650488596522859860, −6.78846332885337771743485298864, −6.14128542502574880260624954719, −5.71893083085973111262398490283, −5.00052336718584334174713723812, −4.18378557007541015174981176590, −2.91841558039042149134097175623, −2.28314095469861550049155237885, −1.57618769335834469782539320356, 0,
1.57618769335834469782539320356, 2.28314095469861550049155237885, 2.91841558039042149134097175623, 4.18378557007541015174981176590, 5.00052336718584334174713723812, 5.71893083085973111262398490283, 6.14128542502574880260624954719, 6.78846332885337771743485298864, 7.83541202961650488596522859860
