| L(s) = 1 | − 687.·2-s − 2.13e4·3-s + 3.41e5·4-s − 1.16e6·5-s + 1.46e7·6-s − 9.37e6·7-s − 1.44e8·8-s + 3.27e8·9-s + 8.04e8·10-s + 8.41e8·11-s − 7.29e9·12-s + 4.06e9·13-s + 6.44e9·14-s + 2.50e10·15-s + 5.46e10·16-s + 1.44e10·17-s − 2.25e11·18-s − 2.26e9·19-s − 3.99e11·20-s + 2.00e11·21-s − 5.78e11·22-s − 2.37e11·23-s + 3.08e12·24-s + 6.05e11·25-s − 2.79e12·26-s − 4.24e12·27-s − 3.20e12·28-s + ⋯ |
| L(s) = 1 | − 1.89·2-s − 1.88·3-s + 2.60·4-s − 1.33·5-s + 3.57·6-s − 0.614·7-s − 3.04·8-s + 2.53·9-s + 2.54·10-s + 1.18·11-s − 4.89·12-s + 1.38·13-s + 1.16·14-s + 2.51·15-s + 3.17·16-s + 0.501·17-s − 4.81·18-s − 0.0305·19-s − 3.48·20-s + 1.15·21-s − 2.24·22-s − 0.631·23-s + 5.72·24-s + 0.794·25-s − 2.62·26-s − 2.89·27-s − 1.60·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(0.2976010878\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2976010878\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 - 5.00e11T \) |
| good | 2 | \( 1 + 687.T + 1.31e5T^{2} \) |
| 3 | \( 1 + 2.13e4T + 1.29e8T^{2} \) |
| 5 | \( 1 + 1.16e6T + 7.62e11T^{2} \) |
| 7 | \( 1 + 9.37e6T + 2.32e14T^{2} \) |
| 11 | \( 1 - 8.41e8T + 5.05e17T^{2} \) |
| 13 | \( 1 - 4.06e9T + 8.65e18T^{2} \) |
| 17 | \( 1 - 1.44e10T + 8.27e20T^{2} \) |
| 19 | \( 1 + 2.26e9T + 5.48e21T^{2} \) |
| 23 | \( 1 + 2.37e11T + 1.41e23T^{2} \) |
| 31 | \( 1 - 7.30e12T + 2.25e25T^{2} \) |
| 37 | \( 1 - 1.73e12T + 4.56e26T^{2} \) |
| 41 | \( 1 + 2.43e13T + 2.61e27T^{2} \) |
| 43 | \( 1 - 8.01e13T + 5.87e27T^{2} \) |
| 47 | \( 1 - 9.47e13T + 2.66e28T^{2} \) |
| 53 | \( 1 + 5.06e14T + 2.05e29T^{2} \) |
| 59 | \( 1 - 1.89e15T + 1.27e30T^{2} \) |
| 61 | \( 1 - 5.95e14T + 2.24e30T^{2} \) |
| 67 | \( 1 - 7.78e14T + 1.10e31T^{2} \) |
| 71 | \( 1 + 2.17e15T + 2.96e31T^{2} \) |
| 73 | \( 1 - 1.29e16T + 4.74e31T^{2} \) |
| 79 | \( 1 - 1.99e16T + 1.81e32T^{2} \) |
| 83 | \( 1 + 2.62e16T + 4.21e32T^{2} \) |
| 89 | \( 1 + 3.29e15T + 1.37e33T^{2} \) |
| 97 | \( 1 + 5.04e16T + 5.95e33T^{2} \) |
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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
−12.25940039003946467305472462458, −11.60872521580043853098744702152, −10.86541641468212196409386751818, −9.716498672599012343388179043325, −8.170132208507554277732135411260, −6.85102773087983214522161094889, −6.15151670093812082149306281121, −3.83853606804548913391173091367, −1.20183801054753012196095268175, −0.53900941982851237518251216268,
0.53900941982851237518251216268, 1.20183801054753012196095268175, 3.83853606804548913391173091367, 6.15151670093812082149306281121, 6.85102773087983214522161094889, 8.170132208507554277732135411260, 9.716498672599012343388179043325, 10.86541641468212196409386751818, 11.60872521580043853098744702152, 12.25940039003946467305472462458
