| L(s) = 1 | + 0.273·2-s − 1.92·4-s − 3.38·5-s + 1.63·7-s − 1.07·8-s − 0.925·10-s − 3.66·11-s − 2.72·13-s + 0.445·14-s + 3.55·16-s − 7.08·17-s + 7.48·19-s + 6.52·20-s − 1.00·22-s − 1.90·23-s + 6.48·25-s − 0.743·26-s − 3.14·28-s − 6.93·29-s + 3.11·32-s − 1.93·34-s − 5.53·35-s − 2.20·37-s + 2.04·38-s + 3.63·40-s − 9.62·41-s − 1.30·43-s + ⋯ |
| L(s) = 1 | + 0.193·2-s − 0.962·4-s − 1.51·5-s + 0.617·7-s − 0.379·8-s − 0.292·10-s − 1.10·11-s − 0.755·13-s + 0.119·14-s + 0.889·16-s − 1.71·17-s + 1.71·19-s + 1.45·20-s − 0.213·22-s − 0.396·23-s + 1.29·25-s − 0.145·26-s − 0.594·28-s − 1.28·29-s + 0.550·32-s − 0.331·34-s − 0.935·35-s − 0.362·37-s + 0.331·38-s + 0.574·40-s − 1.50·41-s − 0.199·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1374543832\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1374543832\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 31 | \( 1 \) |
| good | 2 | \( 1 - 0.273T + 2T^{2} \) |
| 5 | \( 1 + 3.38T + 5T^{2} \) |
| 7 | \( 1 - 1.63T + 7T^{2} \) |
| 11 | \( 1 + 3.66T + 11T^{2} \) |
| 13 | \( 1 + 2.72T + 13T^{2} \) |
| 17 | \( 1 + 7.08T + 17T^{2} \) |
| 19 | \( 1 - 7.48T + 19T^{2} \) |
| 23 | \( 1 + 1.90T + 23T^{2} \) |
| 29 | \( 1 + 6.93T + 29T^{2} \) |
| 37 | \( 1 + 2.20T + 37T^{2} \) |
| 41 | \( 1 + 9.62T + 41T^{2} \) |
| 43 | \( 1 + 1.30T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 + 6.30T + 53T^{2} \) |
| 59 | \( 1 - 4.48T + 59T^{2} \) |
| 61 | \( 1 + 7.37T + 61T^{2} \) |
| 67 | \( 1 + 7.70T + 67T^{2} \) |
| 71 | \( 1 - 4.48T + 71T^{2} \) |
| 73 | \( 1 + 10.6T + 73T^{2} \) |
| 79 | \( 1 + 4.92T + 79T^{2} \) |
| 83 | \( 1 + 6.33T + 83T^{2} \) |
| 89 | \( 1 - 15.6T + 89T^{2} \) |
| 97 | \( 1 - 9.33T + 97T^{2} \) |
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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
−7.61981290863790595078231818329, −7.53003591792475600041101299540, −6.45078311513557060449667736820, −5.25246749703930021634906038582, −4.96544302136914209239223378382, −4.36927982992647587362262645823, −3.55418391026121063694904436453, −2.94532553977764910455210095799, −1.68971759515234621173682778678, −0.17269089214027759919726364275,
0.17269089214027759919726364275, 1.68971759515234621173682778678, 2.94532553977764910455210095799, 3.55418391026121063694904436453, 4.36927982992647587362262645823, 4.96544302136914209239223378382, 5.25246749703930021634906038582, 6.45078311513557060449667736820, 7.53003591792475600041101299540, 7.61981290863790595078231818329
