| L(s) = 1 | − 5.30e4·3-s + 2.95e7·5-s − 3.32e8·7-s − 7.65e9·9-s + 6.27e10·11-s + 1.52e11·13-s − 1.56e12·15-s − 1.24e12·17-s + 1.80e13·19-s + 1.76e13·21-s − 3.13e14·23-s + 3.98e14·25-s + 9.60e14·27-s − 5.19e14·29-s + 6.73e15·31-s − 3.32e15·33-s − 9.84e15·35-s − 5.29e16·37-s − 8.09e15·39-s − 4.78e16·41-s + 6.97e16·43-s − 2.26e17·45-s − 2.25e17·47-s − 4.47e17·49-s + 6.61e16·51-s + 1.23e17·53-s + 1.85e18·55-s + ⋯ |
| L(s) = 1 | − 0.518·3-s + 1.35·5-s − 0.445·7-s − 0.731·9-s + 0.728·11-s + 0.307·13-s − 0.702·15-s − 0.150·17-s + 0.676·19-s + 0.230·21-s − 1.57·23-s + 0.834·25-s + 0.897·27-s − 0.229·29-s + 1.47·31-s − 0.377·33-s − 0.603·35-s − 1.81·37-s − 0.159·39-s − 0.557·41-s + 0.492·43-s − 0.990·45-s − 0.625·47-s − 0.801·49-s + 0.0778·51-s + 0.0971·53-s + 0.987·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 5.30e4T + 1.04e10T^{2} \) |
| 5 | \( 1 - 2.95e7T + 4.76e14T^{2} \) |
| 7 | \( 1 + 3.32e8T + 5.58e17T^{2} \) |
| 11 | \( 1 - 6.27e10T + 7.40e21T^{2} \) |
| 13 | \( 1 - 1.52e11T + 2.47e23T^{2} \) |
| 17 | \( 1 + 1.24e12T + 6.90e25T^{2} \) |
| 19 | \( 1 - 1.80e13T + 7.14e26T^{2} \) |
| 23 | \( 1 + 3.13e14T + 3.94e28T^{2} \) |
| 29 | \( 1 + 5.19e14T + 5.13e30T^{2} \) |
| 31 | \( 1 - 6.73e15T + 2.08e31T^{2} \) |
| 37 | \( 1 + 5.29e16T + 8.55e32T^{2} \) |
| 41 | \( 1 + 4.78e16T + 7.38e33T^{2} \) |
| 43 | \( 1 - 6.97e16T + 2.00e34T^{2} \) |
| 47 | \( 1 + 2.25e17T + 1.30e35T^{2} \) |
| 53 | \( 1 - 1.23e17T + 1.62e36T^{2} \) |
| 59 | \( 1 + 2.43e18T + 1.54e37T^{2} \) |
| 61 | \( 1 - 1.66e18T + 3.10e37T^{2} \) |
| 67 | \( 1 - 2.07e19T + 2.22e38T^{2} \) |
| 71 | \( 1 - 2.24e18T + 7.52e38T^{2} \) |
| 73 | \( 1 - 3.73e19T + 1.34e39T^{2} \) |
| 79 | \( 1 - 1.44e20T + 7.08e39T^{2} \) |
| 83 | \( 1 + 1.23e20T + 1.99e40T^{2} \) |
| 89 | \( 1 + 5.06e20T + 8.65e40T^{2} \) |
| 97 | \( 1 - 9.26e18T + 5.27e41T^{2} \) |
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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
−10.24115141262789073098248014612, −9.488748784026396045638272012189, −8.364678149151337386077990242570, −6.61771845463020771022914391918, −6.02762827923372835719473538861, −5.09593907943540597744841820673, −3.53690328312061907821252249612, −2.29760919716800442629404400689, −1.25249229820107186459653204770, 0,
1.25249229820107186459653204770, 2.29760919716800442629404400689, 3.53690328312061907821252249612, 5.09593907943540597744841820673, 6.02762827923372835719473538861, 6.61771845463020771022914391918, 8.364678149151337386077990242570, 9.488748784026396045638272012189, 10.24115141262789073098248014612
