| L(s) = 1 | − 300.·2-s − 6.56e3·3-s − 4.06e4·4-s + 6.85e5·5-s + 1.97e6·6-s + 5.76e6·7-s + 5.16e7·8-s + 4.30e7·9-s − 2.06e8·10-s − 4.88e8·11-s + 2.66e8·12-s + 6.04e8·13-s − 1.73e9·14-s − 4.49e9·15-s − 1.02e10·16-s − 3.25e10·17-s − 1.29e10·18-s + 2.87e10·19-s − 2.78e10·20-s − 3.78e10·21-s + 1.47e11·22-s − 7.82e10·23-s − 3.38e11·24-s − 2.93e11·25-s − 1.81e11·26-s − 2.82e11·27-s − 2.34e11·28-s + ⋯ |
| L(s) = 1 | − 0.830·2-s − 0.577·3-s − 0.310·4-s + 0.784·5-s + 0.479·6-s + 0.377·7-s + 1.08·8-s + 0.333·9-s − 0.651·10-s − 0.687·11-s + 0.178·12-s + 0.205·13-s − 0.313·14-s − 0.452·15-s − 0.593·16-s − 1.13·17-s − 0.276·18-s + 0.389·19-s − 0.243·20-s − 0.218·21-s + 0.571·22-s − 0.208·23-s − 0.628·24-s − 0.384·25-s − 0.170·26-s − 0.192·27-s − 0.117·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(0.9202505817\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9202505817\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 6.56e3T \) |
| 7 | \( 1 - 5.76e6T \) |
| good | 2 | \( 1 + 300.T + 1.31e5T^{2} \) |
| 5 | \( 1 - 6.85e5T + 7.62e11T^{2} \) |
| 11 | \( 1 + 4.88e8T + 5.05e17T^{2} \) |
| 13 | \( 1 - 6.04e8T + 8.65e18T^{2} \) |
| 17 | \( 1 + 3.25e10T + 8.27e20T^{2} \) |
| 19 | \( 1 - 2.87e10T + 5.48e21T^{2} \) |
| 23 | \( 1 + 7.82e10T + 1.41e23T^{2} \) |
| 29 | \( 1 + 5.82e10T + 7.25e24T^{2} \) |
| 31 | \( 1 + 9.09e11T + 2.25e25T^{2} \) |
| 37 | \( 1 - 1.74e13T + 4.56e26T^{2} \) |
| 41 | \( 1 - 6.09e13T + 2.61e27T^{2} \) |
| 43 | \( 1 - 1.89e13T + 5.87e27T^{2} \) |
| 47 | \( 1 - 2.13e14T + 2.66e28T^{2} \) |
| 53 | \( 1 - 3.07e14T + 2.05e29T^{2} \) |
| 59 | \( 1 + 4.12e14T + 1.27e30T^{2} \) |
| 61 | \( 1 + 1.45e15T + 2.24e30T^{2} \) |
| 67 | \( 1 - 2.09e15T + 1.10e31T^{2} \) |
| 71 | \( 1 - 7.54e15T + 2.96e31T^{2} \) |
| 73 | \( 1 + 8.54e15T + 4.74e31T^{2} \) |
| 79 | \( 1 + 1.91e16T + 1.81e32T^{2} \) |
| 83 | \( 1 + 1.72e15T + 4.21e32T^{2} \) |
| 89 | \( 1 + 3.63e15T + 1.37e33T^{2} \) |
| 97 | \( 1 - 1.04e17T + 5.95e33T^{2} \) |
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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
−13.96354554888632133217646566353, −12.97186841206232958828087058933, −11.15370152845241633861185089068, −10.11019235750343563760837475355, −8.956747886597569513261000427316, −7.52927483515784197191057259291, −5.80922707269571690125430393211, −4.48061978925140764251784554241, −2.05299326739074307950693326591, −0.67467089150812594543348484160,
0.67467089150812594543348484160, 2.05299326739074307950693326591, 4.48061978925140764251784554241, 5.80922707269571690125430393211, 7.52927483515784197191057259291, 8.956747886597569513261000427316, 10.11019235750343563760837475355, 11.15370152845241633861185089068, 12.97186841206232958828087058933, 13.96354554888632133217646566353
