| L(s) = 1 | − 1.85e3·2-s + 1.33e6·4-s − 2.91e7·5-s + 2.00e8·7-s + 1.41e9·8-s + 5.40e10·10-s − 1.09e11·11-s + 8.00e11·13-s − 3.70e11·14-s − 5.41e12·16-s + 8.78e12·17-s − 1.48e13·19-s − 3.89e13·20-s + 2.01e14·22-s − 3.24e14·23-s + 3.75e14·25-s − 1.48e15·26-s + 2.66e14·28-s − 2.05e15·29-s + 1.38e15·31-s + 7.06e15·32-s − 1.62e16·34-s − 5.84e15·35-s + 9.14e15·37-s + 2.74e16·38-s − 4.12e16·40-s − 5.49e16·41-s + ⋯ |
| L(s) = 1 | − 1.27·2-s + 0.635·4-s − 1.33·5-s + 0.267·7-s + 0.465·8-s + 1.71·10-s − 1.26·11-s + 1.60·13-s − 0.342·14-s − 1.23·16-s + 1.05·17-s − 0.555·19-s − 0.850·20-s + 1.62·22-s − 1.63·23-s + 0.787·25-s − 2.05·26-s + 0.170·28-s − 0.908·29-s + 0.303·31-s + 1.10·32-s − 1.35·34-s − 0.357·35-s + 0.312·37-s + 0.710·38-s − 0.622·40-s − 0.639·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + 1.85e3T + 2.09e6T^{2} \) |
| 5 | \( 1 + 2.91e7T + 4.76e14T^{2} \) |
| 7 | \( 1 - 2.00e8T + 5.58e17T^{2} \) |
| 11 | \( 1 + 1.09e11T + 7.40e21T^{2} \) |
| 13 | \( 1 - 8.00e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 8.78e12T + 6.90e25T^{2} \) |
| 19 | \( 1 + 1.48e13T + 7.14e26T^{2} \) |
| 23 | \( 1 + 3.24e14T + 3.94e28T^{2} \) |
| 29 | \( 1 + 2.05e15T + 5.13e30T^{2} \) |
| 31 | \( 1 - 1.38e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 9.14e15T + 8.55e32T^{2} \) |
| 41 | \( 1 + 5.49e16T + 7.38e33T^{2} \) |
| 43 | \( 1 - 3.84e16T + 2.00e34T^{2} \) |
| 47 | \( 1 - 5.87e17T + 1.30e35T^{2} \) |
| 53 | \( 1 + 5.47e17T + 1.62e36T^{2} \) |
| 59 | \( 1 - 5.55e16T + 1.54e37T^{2} \) |
| 61 | \( 1 - 1.71e18T + 3.10e37T^{2} \) |
| 67 | \( 1 - 2.59e16T + 2.22e38T^{2} \) |
| 71 | \( 1 - 1.92e19T + 7.52e38T^{2} \) |
| 73 | \( 1 + 4.51e19T + 1.34e39T^{2} \) |
| 79 | \( 1 - 1.06e20T + 7.08e39T^{2} \) |
| 83 | \( 1 + 2.32e20T + 1.99e40T^{2} \) |
| 89 | \( 1 + 4.71e20T + 8.65e40T^{2} \) |
| 97 | \( 1 - 6.71e20T + 5.27e41T^{2} \) |
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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
−10.02106734465887974121463990461, −8.596137632549100668490622536941, −8.049106386493016699081556864119, −7.43240721503761062101130229281, −5.86478608920510252606707578729, −4.39963969239875297706194393701, −3.46295573028805755276607931922, −1.94075292217226474265432738686, −0.801818681266095791954270353962, 0,
0.801818681266095791954270353962, 1.94075292217226474265432738686, 3.46295573028805755276607931922, 4.39963969239875297706194393701, 5.86478608920510252606707578729, 7.43240721503761062101130229281, 8.049106386493016699081556864119, 8.596137632549100668490622536941, 10.02106734465887974121463990461
