| L(s) = 1 | + 2.56e3·2-s + 4.50e6·4-s − 2.83e7·5-s + 6.46e8·7-s + 6.18e9·8-s − 7.27e10·10-s − 1.38e11·11-s + 7.58e11·13-s + 1.66e12·14-s + 6.44e12·16-s − 7.31e12·17-s + 5.05e12·19-s − 1.27e14·20-s − 3.55e14·22-s + 1.51e14·23-s + 3.25e14·25-s + 1.94e15·26-s + 2.91e15·28-s + 1.38e15·29-s − 9.69e14·31-s + 3.58e15·32-s − 1.87e16·34-s − 1.83e16·35-s − 3.45e16·37-s + 1.29e16·38-s − 1.75e17·40-s − 6.00e16·41-s + ⋯ |
| L(s) = 1 | + 1.77·2-s + 2.14·4-s − 1.29·5-s + 0.865·7-s + 2.03·8-s − 2.30·10-s − 1.60·11-s + 1.52·13-s + 1.53·14-s + 1.46·16-s − 0.880·17-s + 0.189·19-s − 2.78·20-s − 2.85·22-s + 0.761·23-s + 0.683·25-s + 2.70·26-s + 1.85·28-s + 0.613·29-s − 0.212·31-s + 0.563·32-s − 1.56·34-s − 1.12·35-s − 1.18·37-s + 0.335·38-s − 2.64·40-s − 0.698·41-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 | \( 1 - 2.56e3T + 2.09e6T^{2} \) |
| 5 | \( 1 + 2.83e7T + 4.76e14T^{2} \) |
| 7 | \( 1 - 6.46e8T + 5.58e17T^{2} \) |
| 11 | \( 1 + 1.38e11T + 7.40e21T^{2} \) |
| 13 | \( 1 - 7.58e11T + 2.47e23T^{2} \) |
| 17 | \( 1 + 7.31e12T + 6.90e25T^{2} \) |
| 19 | \( 1 - 5.05e12T + 7.14e26T^{2} \) |
| 23 | \( 1 - 1.51e14T + 3.94e28T^{2} \) |
| 29 | \( 1 - 1.38e15T + 5.13e30T^{2} \) |
| 31 | \( 1 + 9.69e14T + 2.08e31T^{2} \) |
| 37 | \( 1 + 3.45e16T + 8.55e32T^{2} \) |
| 41 | \( 1 + 6.00e16T + 7.38e33T^{2} \) |
| 43 | \( 1 - 1.43e17T + 2.00e34T^{2} \) |
| 47 | \( 1 + 4.45e17T + 1.30e35T^{2} \) |
| 53 | \( 1 + 1.71e18T + 1.62e36T^{2} \) |
| 59 | \( 1 + 7.82e17T + 1.54e37T^{2} \) |
| 61 | \( 1 + 5.38e18T + 3.10e37T^{2} \) |
| 67 | \( 1 - 7.75e18T + 2.22e38T^{2} \) |
| 71 | \( 1 + 8.59e18T + 7.52e38T^{2} \) |
| 73 | \( 1 - 1.19e19T + 1.34e39T^{2} \) |
| 79 | \( 1 - 2.74e19T + 7.08e39T^{2} \) |
| 83 | \( 1 - 1.20e20T + 1.99e40T^{2} \) |
| 89 | \( 1 + 5.26e20T + 8.65e40T^{2} \) |
| 97 | \( 1 - 9.65e20T + 5.27e41T^{2} \) |
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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
−10.92960943716029994355461740738, −8.456644718769838620824913697795, −7.66716472297212556819838181073, −6.54742248443389343997189705163, −5.27209746895284912855069648419, −4.60752273996403818911705039004, −3.65293918007343650533521745498, −2.81964238436448074348282651720, −1.54178961824202162628802405902, 0,
1.54178961824202162628802405902, 2.81964238436448074348282651720, 3.65293918007343650533521745498, 4.60752273996403818911705039004, 5.27209746895284912855069648419, 6.54742248443389343997189705163, 7.66716472297212556819838181073, 8.456644718769838620824913697795, 10.92960943716029994355461740738
