| L(s) = 1 | + 1.61·5-s + 2.23·7-s − 0.236·13-s + 2.61·17-s − 3.61·19-s − 3.47·23-s − 2.38·25-s − 2·29-s + 3.09·31-s + 3.61·35-s − 6.70·37-s − 8.23·41-s − 11.9·43-s − 7.38·47-s − 1.99·49-s − 11.5·53-s + 2.85·59-s − 4.38·61-s − 0.381·65-s − 14.0·67-s + 8.85·71-s + 13.2·73-s − 1.76·79-s − 9.18·83-s + 4.23·85-s + 12.2·89-s − 0.527·91-s + ⋯ |
| L(s) = 1 | + 0.723·5-s + 0.845·7-s − 0.0654·13-s + 0.634·17-s − 0.830·19-s − 0.723·23-s − 0.476·25-s − 0.371·29-s + 0.555·31-s + 0.611·35-s − 1.10·37-s − 1.28·41-s − 1.82·43-s − 1.07·47-s − 0.285·49-s − 1.58·53-s + 0.371·59-s − 0.561·61-s − 0.0473·65-s − 1.72·67-s + 1.05·71-s + 1.54·73-s − 0.198·79-s − 1.00·83-s + 0.459·85-s + 1.29·89-s − 0.0553·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 1.61T + 5T^{2} \) |
| 7 | \( 1 - 2.23T + 7T^{2} \) |
| 13 | \( 1 + 0.236T + 13T^{2} \) |
| 17 | \( 1 - 2.61T + 17T^{2} \) |
| 19 | \( 1 + 3.61T + 19T^{2} \) |
| 23 | \( 1 + 3.47T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 3.09T + 31T^{2} \) |
| 37 | \( 1 + 6.70T + 37T^{2} \) |
| 41 | \( 1 + 8.23T + 41T^{2} \) |
| 43 | \( 1 + 11.9T + 43T^{2} \) |
| 47 | \( 1 + 7.38T + 47T^{2} \) |
| 53 | \( 1 + 11.5T + 53T^{2} \) |
| 59 | \( 1 - 2.85T + 59T^{2} \) |
| 61 | \( 1 + 4.38T + 61T^{2} \) |
| 67 | \( 1 + 14.0T + 67T^{2} \) |
| 71 | \( 1 - 8.85T + 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 + 1.76T + 79T^{2} \) |
| 83 | \( 1 + 9.18T + 83T^{2} \) |
| 89 | \( 1 - 12.2T + 89T^{2} \) |
| 97 | \( 1 + 5.90T + 97T^{2} \) |
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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
−7.51282923381390305345660141428, −6.56861629026406530951028805120, −6.14755225938855360163943133845, −5.19217975306917314304185703046, −4.85965216463888448430745825808, −3.84618222110123475021757807381, −3.06870265427677379463644670704, −1.88742224765352344311867975743, −1.61183006665976634318827993613, 0,
1.61183006665976634318827993613, 1.88742224765352344311867975743, 3.06870265427677379463644670704, 3.84618222110123475021757807381, 4.85965216463888448430745825808, 5.19217975306917314304185703046, 6.14755225938855360163943133845, 6.56861629026406530951028805120, 7.51282923381390305345660141428
