| L(s) = 1 | + 3.39e7·5-s − 5.28e8·7-s + 1.21e11·11-s + 4.33e11·13-s + 1.31e13·17-s + 2.18e13·19-s − 1.40e14·23-s + 6.74e14·25-s − 1.17e15·29-s + 9.57e14·31-s − 1.79e16·35-s − 3.53e16·37-s + 1.71e17·41-s + 1.35e17·43-s + 5.75e17·47-s − 2.78e17·49-s + 9.62e17·53-s + 4.13e18·55-s − 4.87e18·59-s + 4.59e18·61-s + 1.47e19·65-s − 1.78e19·67-s − 2.93e19·71-s − 8.32e18·73-s − 6.44e19·77-s − 7.05e19·79-s − 2.11e20·83-s + ⋯ |
| L(s) = 1 | + 1.55·5-s − 0.707·7-s + 1.41·11-s + 0.872·13-s + 1.58·17-s + 0.818·19-s − 0.708·23-s + 1.41·25-s − 0.516·29-s + 0.209·31-s − 1.09·35-s − 1.20·37-s + 1.99·41-s + 0.957·43-s + 1.59·47-s − 0.499·49-s + 0.755·53-s + 2.20·55-s − 1.24·59-s + 0.824·61-s + 1.35·65-s − 1.19·67-s − 1.07·71-s − 0.226·73-s − 1.00·77-s − 0.838·79-s − 1.49·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(4.135991361\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.135991361\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 3.39e7T + 4.76e14T^{2} \) |
| 7 | \( 1 + 5.28e8T + 5.58e17T^{2} \) |
| 11 | \( 1 - 1.21e11T + 7.40e21T^{2} \) |
| 13 | \( 1 - 4.33e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 1.31e13T + 6.90e25T^{2} \) |
| 19 | \( 1 - 2.18e13T + 7.14e26T^{2} \) |
| 23 | \( 1 + 1.40e14T + 3.94e28T^{2} \) |
| 29 | \( 1 + 1.17e15T + 5.13e30T^{2} \) |
| 31 | \( 1 - 9.57e14T + 2.08e31T^{2} \) |
| 37 | \( 1 + 3.53e16T + 8.55e32T^{2} \) |
| 41 | \( 1 - 1.71e17T + 7.38e33T^{2} \) |
| 43 | \( 1 - 1.35e17T + 2.00e34T^{2} \) |
| 47 | \( 1 - 5.75e17T + 1.30e35T^{2} \) |
| 53 | \( 1 - 9.62e17T + 1.62e36T^{2} \) |
| 59 | \( 1 + 4.87e18T + 1.54e37T^{2} \) |
| 61 | \( 1 - 4.59e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 1.78e19T + 2.22e38T^{2} \) |
| 71 | \( 1 + 2.93e19T + 7.52e38T^{2} \) |
| 73 | \( 1 + 8.32e18T + 1.34e39T^{2} \) |
| 79 | \( 1 + 7.05e19T + 7.08e39T^{2} \) |
| 83 | \( 1 + 2.11e20T + 1.99e40T^{2} \) |
| 89 | \( 1 + 2.62e20T + 8.65e40T^{2} \) |
| 97 | \( 1 - 3.84e20T + 5.27e41T^{2} \) |
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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
−10.43201911827748748566686058204, −9.593671338914522943817256971977, −8.953832552545590040366023283993, −7.30296757660894795095755671342, −6.04635110705038371080546240891, −5.72534826772087252621340762939, −3.97261159556828709571603902990, −2.93544947670646895612947363744, −1.61275167347292751596364029617, −0.943573727678738474526503437332,
0.943573727678738474526503437332, 1.61275167347292751596364029617, 2.93544947670646895612947363744, 3.97261159556828709571603902990, 5.72534826772087252621340762939, 6.04635110705038371080546240891, 7.30296757660894795095755671342, 8.953832552545590040366023283993, 9.593671338914522943817256971977, 10.43201911827748748566686058204
