| L(s) = 1 | − 2.78e3·2-s + 5.67e6·4-s + 9.76e6·5-s − 1.37e9·7-s − 9.96e9·8-s − 2.72e10·10-s − 2.99e10·11-s − 7.63e11·13-s + 3.84e12·14-s + 1.58e13·16-s + 1.14e13·17-s − 5.19e12·19-s + 5.53e13·20-s + 8.35e13·22-s + 8.03e13·23-s + 9.53e13·25-s + 2.12e15·26-s − 7.82e15·28-s + 2.01e14·29-s + 1.53e15·31-s − 2.33e16·32-s − 3.20e16·34-s − 1.34e16·35-s + 1.53e16·37-s + 1.44e16·38-s − 9.72e16·40-s + 7.88e16·41-s + ⋯ |
| L(s) = 1 | − 1.92·2-s + 2.70·4-s + 0.447·5-s − 1.84·7-s − 3.28·8-s − 0.860·10-s − 0.348·11-s − 1.53·13-s + 3.55·14-s + 3.60·16-s + 1.38·17-s − 0.194·19-s + 1.20·20-s + 0.670·22-s + 0.404·23-s + 0.199·25-s + 2.95·26-s − 4.99·28-s + 0.0887·29-s + 0.336·31-s − 3.66·32-s − 2.66·34-s − 0.825·35-s + 0.526·37-s + 0.374·38-s − 1.46·40-s + 0.917·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - 9.76e6T \) |
| good | 2 | \( 1 + 2.78e3T + 2.09e6T^{2} \) |
| 7 | \( 1 + 1.37e9T + 5.58e17T^{2} \) |
| 11 | \( 1 + 2.99e10T + 7.40e21T^{2} \) |
| 13 | \( 1 + 7.63e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 1.14e13T + 6.90e25T^{2} \) |
| 19 | \( 1 + 5.19e12T + 7.14e26T^{2} \) |
| 23 | \( 1 - 8.03e13T + 3.94e28T^{2} \) |
| 29 | \( 1 - 2.01e14T + 5.13e30T^{2} \) |
| 31 | \( 1 - 1.53e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 1.53e16T + 8.55e32T^{2} \) |
| 41 | \( 1 - 7.88e16T + 7.38e33T^{2} \) |
| 43 | \( 1 + 1.61e17T + 2.00e34T^{2} \) |
| 47 | \( 1 - 3.76e17T + 1.30e35T^{2} \) |
| 53 | \( 1 + 3.24e17T + 1.62e36T^{2} \) |
| 59 | \( 1 - 9.88e17T + 1.54e37T^{2} \) |
| 61 | \( 1 - 1.01e19T + 3.10e37T^{2} \) |
| 67 | \( 1 + 1.02e19T + 2.22e38T^{2} \) |
| 71 | \( 1 + 3.23e19T + 7.52e38T^{2} \) |
| 73 | \( 1 + 2.89e19T + 1.34e39T^{2} \) |
| 79 | \( 1 - 3.51e19T + 7.08e39T^{2} \) |
| 83 | \( 1 + 2.86e19T + 1.99e40T^{2} \) |
| 89 | \( 1 + 5.20e19T + 8.65e40T^{2} \) |
| 97 | \( 1 - 7.90e20T + 5.27e41T^{2} \) |
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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
−10.26988660738847740628860542284, −9.899884873575276472539749569648, −9.067971885225474776581451569885, −7.63746092975610879620538554155, −6.79520032878164819329314407129, −5.74297014885517462650724030174, −3.10712285686303602807190233504, −2.38129273675298422181362270363, −0.878809702750229247828947995340, 0,
0.878809702750229247828947995340, 2.38129273675298422181362270363, 3.10712285686303602807190233504, 5.74297014885517462650724030174, 6.79520032878164819329314407129, 7.63746092975610879620538554155, 9.067971885225474776581451569885, 9.899884873575276472539749569648, 10.26988660738847740628860542284
