| L(s) = 1 | + 1.25e3·2-s − 5.27e5·4-s − 3.03e7·5-s − 1.16e9·7-s − 3.28e9·8-s − 3.80e10·10-s − 1.56e10·11-s + 7.23e11·13-s − 1.46e12·14-s − 3.01e12·16-s − 2.77e11·17-s + 4.85e13·19-s + 1.60e13·20-s − 1.96e13·22-s − 1.23e14·23-s + 4.45e14·25-s + 9.06e14·26-s + 6.15e14·28-s + 1.02e15·29-s − 1.46e15·31-s + 3.12e15·32-s − 3.47e14·34-s + 3.54e16·35-s + 4.55e16·37-s + 6.08e16·38-s + 9.98e16·40-s − 7.41e16·41-s + ⋯ |
| L(s) = 1 | + 0.865·2-s − 0.251·4-s − 1.39·5-s − 1.56·7-s − 1.08·8-s − 1.20·10-s − 0.182·11-s + 1.45·13-s − 1.34·14-s − 0.685·16-s − 0.0334·17-s + 1.81·19-s + 0.349·20-s − 0.157·22-s − 0.622·23-s + 0.934·25-s + 1.25·26-s + 0.392·28-s + 0.453·29-s − 0.320·31-s + 0.490·32-s − 0.0289·34-s + 2.17·35-s + 1.55·37-s + 1.57·38-s + 1.50·40-s − 0.862·41-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 | \( 1 - 1.25e3T + 2.09e6T^{2} \) |
| 5 | \( 1 + 3.03e7T + 4.76e14T^{2} \) |
| 7 | \( 1 + 1.16e9T + 5.58e17T^{2} \) |
| 11 | \( 1 + 1.56e10T + 7.40e21T^{2} \) |
| 13 | \( 1 - 7.23e11T + 2.47e23T^{2} \) |
| 17 | \( 1 + 2.77e11T + 6.90e25T^{2} \) |
| 19 | \( 1 - 4.85e13T + 7.14e26T^{2} \) |
| 23 | \( 1 + 1.23e14T + 3.94e28T^{2} \) |
| 29 | \( 1 - 1.02e15T + 5.13e30T^{2} \) |
| 31 | \( 1 + 1.46e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 4.55e16T + 8.55e32T^{2} \) |
| 41 | \( 1 + 7.41e16T + 7.38e33T^{2} \) |
| 43 | \( 1 - 1.63e16T + 2.00e34T^{2} \) |
| 47 | \( 1 + 4.43e17T + 1.30e35T^{2} \) |
| 53 | \( 1 - 1.56e18T + 1.62e36T^{2} \) |
| 59 | \( 1 + 6.07e18T + 1.54e37T^{2} \) |
| 61 | \( 1 - 4.24e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 3.06e18T + 2.22e38T^{2} \) |
| 71 | \( 1 + 2.92e18T + 7.52e38T^{2} \) |
| 73 | \( 1 + 4.61e19T + 1.34e39T^{2} \) |
| 79 | \( 1 + 1.12e19T + 7.08e39T^{2} \) |
| 83 | \( 1 - 1.22e19T + 1.99e40T^{2} \) |
| 89 | \( 1 - 4.21e20T + 8.65e40T^{2} \) |
| 97 | \( 1 + 1.20e20T + 5.27e41T^{2} \) |
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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
−9.928581258398043802624797841657, −8.913562172480127826693934704892, −7.80903115979337988747156746247, −6.55963711923483160745794473904, −5.64876680022955500011976162481, −4.31005001956979238384785959797, −3.48165306848037183198336486230, −3.07077568597965486405789385102, −0.864974456706238540213095520037, 0,
0.864974456706238540213095520037, 3.07077568597965486405789385102, 3.48165306848037183198336486230, 4.31005001956979238384785959797, 5.64876680022955500011976162481, 6.55963711923483160745794473904, 7.80903115979337988747156746247, 8.913562172480127826693934704892, 9.928581258398043802624797841657
