| L(s) = 1 | − 8.19e3·2-s + 1.86e5·3-s + 6.71e7·4-s + 1.22e9·5-s − 1.53e9·6-s + 1.52e11·7-s − 5.49e11·8-s − 7.59e12·9-s − 1.00e13·10-s + 5.20e13·11-s + 1.25e13·12-s − 9.66e14·13-s − 1.24e15·14-s + 2.28e14·15-s + 4.50e15·16-s − 8.84e15·17-s + 6.21e16·18-s − 4.96e16·19-s + 8.19e16·20-s + 2.84e16·21-s − 4.26e17·22-s + 2.39e18·23-s − 1.02e17·24-s + 1.49e18·25-s + 7.91e18·26-s − 2.84e18·27-s + 1.02e19·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.0676·3-s + 0.5·4-s + 0.447·5-s − 0.0478·6-s + 0.593·7-s − 0.353·8-s − 0.995·9-s − 0.316·10-s + 0.454·11-s + 0.0338·12-s − 0.885·13-s − 0.419·14-s + 0.0302·15-s + 0.250·16-s − 0.216·17-s + 0.703·18-s − 0.270·19-s + 0.223·20-s + 0.0401·21-s − 0.321·22-s + 0.990·23-s − 0.0239·24-s + 0.199·25-s + 0.625·26-s − 0.134·27-s + 0.296·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(28-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+27/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(14)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{29}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8.19e3T \) |
| 5 | \( 1 - 1.22e9T \) |
| good | 3 | \( 1 - 1.86e5T + 7.62e12T^{2} \) |
| 7 | \( 1 - 1.52e11T + 6.57e22T^{2} \) |
| 11 | \( 1 - 5.20e13T + 1.31e28T^{2} \) |
| 13 | \( 1 + 9.66e14T + 1.19e30T^{2} \) |
| 17 | \( 1 + 8.84e15T + 1.66e33T^{2} \) |
| 19 | \( 1 + 4.96e16T + 3.36e34T^{2} \) |
| 23 | \( 1 - 2.39e18T + 5.84e36T^{2} \) |
| 29 | \( 1 - 5.37e18T + 3.05e39T^{2} \) |
| 31 | \( 1 - 1.60e20T + 1.84e40T^{2} \) |
| 37 | \( 1 + 2.24e21T + 2.19e42T^{2} \) |
| 41 | \( 1 + 4.36e21T + 3.50e43T^{2} \) |
| 43 | \( 1 + 5.19e21T + 1.26e44T^{2} \) |
| 47 | \( 1 - 1.42e21T + 1.40e45T^{2} \) |
| 53 | \( 1 + 2.24e23T + 3.59e46T^{2} \) |
| 59 | \( 1 + 1.17e24T + 6.50e47T^{2} \) |
| 61 | \( 1 + 2.05e24T + 1.59e48T^{2} \) |
| 67 | \( 1 - 3.81e24T + 2.01e49T^{2} \) |
| 71 | \( 1 + 3.12e24T + 9.63e49T^{2} \) |
| 73 | \( 1 - 1.49e25T + 2.04e50T^{2} \) |
| 79 | \( 1 - 1.41e25T + 1.72e51T^{2} \) |
| 83 | \( 1 + 7.95e24T + 6.53e51T^{2} \) |
| 89 | \( 1 + 1.05e26T + 4.30e52T^{2} \) |
| 97 | \( 1 + 2.67e26T + 4.39e53T^{2} \) |
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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
−13.97756821922551499166121253593, −12.07522271487377906900866341107, −10.82351813602922164737248597228, −9.349324796154985816624834683360, −8.190493865173437379062652528635, −6.61010164811198320189881389167, −5.03604025707529726661101869382, −2.88574255606246008863666505496, −1.56573529971417964589857727623, 0,
1.56573529971417964589857727623, 2.88574255606246008863666505496, 5.03604025707529726661101869382, 6.61010164811198320189881389167, 8.190493865173437379062652528635, 9.349324796154985816624834683360, 10.82351813602922164737248597228, 12.07522271487377906900866341107, 13.97756821922551499166121253593
