| L(s) = 1 | − 1.01·3-s − 1.22·5-s − 7-s − 1.97·9-s − 4.20·11-s + 2.85·13-s + 1.24·15-s − 0.264·17-s + 6.41·19-s + 1.01·21-s − 1.54·23-s − 3.49·25-s + 5.03·27-s − 0.464·29-s + 6.04·31-s + 4.25·33-s + 1.22·35-s − 9.40·37-s − 2.88·39-s + 11.0·41-s + 4.78·43-s + 2.42·45-s − 3.12·47-s + 49-s + 0.267·51-s + 0.608·53-s + 5.16·55-s + ⋯ |
| L(s) = 1 | − 0.584·3-s − 0.548·5-s − 0.377·7-s − 0.658·9-s − 1.26·11-s + 0.792·13-s + 0.320·15-s − 0.0641·17-s + 1.47·19-s + 0.220·21-s − 0.322·23-s − 0.698·25-s + 0.968·27-s − 0.0861·29-s + 1.08·31-s + 0.741·33-s + 0.207·35-s − 1.54·37-s − 0.462·39-s + 1.72·41-s + 0.729·43-s + 0.361·45-s − 0.455·47-s + 0.142·49-s + 0.0374·51-s + 0.0836·53-s + 0.696·55-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 | \( 1 + 1.01T + 3T^{2} \) |
| 5 | \( 1 + 1.22T + 5T^{2} \) |
| 11 | \( 1 + 4.20T + 11T^{2} \) |
| 13 | \( 1 - 2.85T + 13T^{2} \) |
| 17 | \( 1 + 0.264T + 17T^{2} \) |
| 19 | \( 1 - 6.41T + 19T^{2} \) |
| 23 | \( 1 + 1.54T + 23T^{2} \) |
| 29 | \( 1 + 0.464T + 29T^{2} \) |
| 31 | \( 1 - 6.04T + 31T^{2} \) |
| 37 | \( 1 + 9.40T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 - 4.78T + 43T^{2} \) |
| 47 | \( 1 + 3.12T + 47T^{2} \) |
| 53 | \( 1 - 0.608T + 53T^{2} \) |
| 59 | \( 1 - 6.54T + 59T^{2} \) |
| 61 | \( 1 - 6.88T + 61T^{2} \) |
| 67 | \( 1 - 4.72T + 67T^{2} \) |
| 71 | \( 1 - 9.03T + 71T^{2} \) |
| 73 | \( 1 + 14.8T + 73T^{2} \) |
| 79 | \( 1 + 12.5T + 79T^{2} \) |
| 83 | \( 1 - 1.01T + 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 97 | \( 1 + 14.2T + 97T^{2} \) |
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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
−7.59367682817315333032915709297, −6.92371927021624914502489738829, −5.95203762483302432236677431183, −5.60864120481357662868733994737, −4.86873928828350430833571454275, −3.91352215877040366926878564352, −3.15919175343319516078926442799, −2.43347275987055288497431422722, −0.988965875824657211036995876866, 0,
0.988965875824657211036995876866, 2.43347275987055288497431422722, 3.15919175343319516078926442799, 3.91352215877040366926878564352, 4.86873928828350430833571454275, 5.60864120481357662868733994737, 5.95203762483302432236677431183, 6.92371927021624914502489738829, 7.59367682817315333032915709297
