| L(s) = 1 | − 67.9·2-s − 249.·3-s + 2.57e3·4-s + 1.21e4·5-s + 1.69e4·6-s + 7.49e4·7-s − 3.56e4·8-s − 1.15e5·9-s − 8.24e5·10-s + 2.31e5·11-s − 6.41e5·12-s + 2.26e6·13-s − 5.09e6·14-s − 3.02e6·15-s − 2.84e6·16-s − 3.23e6·17-s + 7.81e6·18-s − 1.95e6·19-s + 3.12e7·20-s − 1.86e7·21-s − 1.57e7·22-s + 3.31e7·23-s + 8.89e6·24-s + 9.83e7·25-s − 1.53e8·26-s + 7.28e7·27-s + 1.92e8·28-s + ⋯ |
| L(s) = 1 | − 1.50·2-s − 0.592·3-s + 1.25·4-s + 1.73·5-s + 0.889·6-s + 1.68·7-s − 0.385·8-s − 0.649·9-s − 2.60·10-s + 0.433·11-s − 0.744·12-s + 1.69·13-s − 2.53·14-s − 1.02·15-s − 0.677·16-s − 0.552·17-s + 0.975·18-s − 0.180·19-s + 2.18·20-s − 0.997·21-s − 0.650·22-s + 1.07·23-s + 0.228·24-s + 2.01·25-s − 2.54·26-s + 0.976·27-s + 2.11·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.290208042\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.290208042\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + 2.05e7T \) |
| good | 2 | \( 1 + 67.9T + 2.04e3T^{2} \) |
| 3 | \( 1 + 249.T + 1.77e5T^{2} \) |
| 5 | \( 1 - 1.21e4T + 4.88e7T^{2} \) |
| 7 | \( 1 - 7.49e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 2.31e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 2.26e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + 3.23e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 1.95e6T + 1.16e14T^{2} \) |
| 23 | \( 1 - 3.31e7T + 9.52e14T^{2} \) |
| 31 | \( 1 + 2.26e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 2.64e5T + 1.77e17T^{2} \) |
| 41 | \( 1 - 1.31e7T + 5.50e17T^{2} \) |
| 43 | \( 1 + 5.31e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + 5.10e8T + 2.47e18T^{2} \) |
| 53 | \( 1 + 5.33e9T + 9.26e18T^{2} \) |
| 59 | \( 1 - 1.64e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 1.21e10T + 4.35e19T^{2} \) |
| 67 | \( 1 - 2.79e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 1.61e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 1.29e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 4.82e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 3.38e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 1.85e9T + 2.77e21T^{2} \) |
| 97 | \( 1 + 6.53e10T + 7.15e21T^{2} \) |
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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
−14.56751277627844763713110840939, −13.45531254768450917123610503829, −11.18175316359684832260262268001, −10.82777214305962192326441191387, −9.196673914127384137505604441169, −8.411804422192536745759576720693, −6.51369884982194185946596717495, −5.23511280177832175284512144135, −1.92015759979425527366174244325, −1.10434044309483971952719157582,
1.10434044309483971952719157582, 1.92015759979425527366174244325, 5.23511280177832175284512144135, 6.51369884982194185946596717495, 8.411804422192536745759576720693, 9.196673914127384137505604441169, 10.82777214305962192326441191387, 11.18175316359684832260262268001, 13.45531254768450917123610503829, 14.56751277627844763713110840939
