| L(s) = 1 | + 6.22e4·3-s − 1.24e6·5-s − 2.63e8·7-s − 6.58e9·9-s − 3.37e10·11-s + 2.50e11·13-s − 7.75e10·15-s + 2.34e11·17-s − 3.24e13·19-s − 1.63e13·21-s − 9.63e13·23-s − 4.75e14·25-s − 1.06e15·27-s − 3.86e15·29-s − 2.26e15·31-s − 2.09e15·33-s + 3.27e14·35-s + 4.75e15·37-s + 1.56e16·39-s + 3.56e16·41-s + 2.20e17·43-s + 8.20e15·45-s + 4.79e17·47-s − 4.89e17·49-s + 1.46e16·51-s − 1.86e18·53-s + 4.19e16·55-s + ⋯ |
| L(s) = 1 | + 0.608·3-s − 0.0570·5-s − 0.351·7-s − 0.629·9-s − 0.391·11-s + 0.504·13-s − 0.0347·15-s + 0.0282·17-s − 1.21·19-s − 0.214·21-s − 0.484·23-s − 0.996·25-s − 0.991·27-s − 1.70·29-s − 0.497·31-s − 0.238·33-s + 0.0200·35-s + 0.162·37-s + 0.307·39-s + 0.414·41-s + 1.55·43-s + 0.0359·45-s + 1.33·47-s − 0.876·49-s + 0.0171·51-s − 1.46·53-s + 0.0223·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| good | 3 | \( 1 - 6.22e4T + 1.04e10T^{2} \) |
| 5 | \( 1 + 1.24e6T + 4.76e14T^{2} \) |
| 7 | \( 1 + 2.63e8T + 5.58e17T^{2} \) |
| 11 | \( 1 + 3.37e10T + 7.40e21T^{2} \) |
| 13 | \( 1 - 2.50e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 2.34e11T + 6.90e25T^{2} \) |
| 19 | \( 1 + 3.24e13T + 7.14e26T^{2} \) |
| 23 | \( 1 + 9.63e13T + 3.94e28T^{2} \) |
| 29 | \( 1 + 3.86e15T + 5.13e30T^{2} \) |
| 31 | \( 1 + 2.26e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 4.75e15T + 8.55e32T^{2} \) |
| 41 | \( 1 - 3.56e16T + 7.38e33T^{2} \) |
| 43 | \( 1 - 2.20e17T + 2.00e34T^{2} \) |
| 47 | \( 1 - 4.79e17T + 1.30e35T^{2} \) |
| 53 | \( 1 + 1.86e18T + 1.62e36T^{2} \) |
| 59 | \( 1 - 1.23e18T + 1.54e37T^{2} \) |
| 61 | \( 1 + 6.27e18T + 3.10e37T^{2} \) |
| 67 | \( 1 - 1.72e19T + 2.22e38T^{2} \) |
| 71 | \( 1 - 3.13e19T + 7.52e38T^{2} \) |
| 73 | \( 1 - 5.49e19T + 1.34e39T^{2} \) |
| 79 | \( 1 + 8.34e19T + 7.08e39T^{2} \) |
| 83 | \( 1 + 1.22e20T + 1.99e40T^{2} \) |
| 89 | \( 1 - 1.21e20T + 8.65e40T^{2} \) |
| 97 | \( 1 - 3.73e20T + 5.27e41T^{2} \) |
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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
−15.59661396954442860559235109012, −14.23317716129675432732428877895, −12.87488982868465475853803342660, −11.04503870474994282883890154042, −9.267306333418824615595046301640, −7.88163221860336425749332088018, −5.91823067859093694876717894541, −3.76850654592360856013747804548, −2.20807375115340758973950602148, 0,
2.20807375115340758973950602148, 3.76850654592360856013747804548, 5.91823067859093694876717894541, 7.88163221860336425749332088018, 9.267306333418824615595046301640, 11.04503870474994282883890154042, 12.87488982868465475853803342660, 14.23317716129675432732428877895, 15.59661396954442860559235109012
