| L(s) = 1 | − 3.53·2-s + 13.5·3-s − 19.5·4-s − 25·5-s − 48·6-s − 49·7-s + 181.·8-s − 58.2·9-s + 88.2·10-s − 691.·11-s − 265.·12-s − 502.·13-s + 173.·14-s − 339.·15-s − 17.5·16-s − 991.·17-s + 205.·18-s + 661.·19-s + 488.·20-s − 666.·21-s + 2.44e3·22-s + 3.41e3·23-s + 2.47e3·24-s + 625·25-s + 1.77e3·26-s − 4.09e3·27-s + 957.·28-s + ⋯ |
| L(s) = 1 | − 0.624·2-s + 0.872·3-s − 0.610·4-s − 0.447·5-s − 0.544·6-s − 0.377·7-s + 1.00·8-s − 0.239·9-s + 0.279·10-s − 1.72·11-s − 0.532·12-s − 0.824·13-s + 0.235·14-s − 0.389·15-s − 0.0171·16-s − 0.831·17-s + 0.149·18-s + 0.420·19-s + 0.272·20-s − 0.329·21-s + 1.07·22-s + 1.34·23-s + 0.876·24-s + 0.200·25-s + 0.514·26-s − 1.08·27-s + 0.230·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 7 | \( 1 + 49T \) |
| good | 2 | \( 1 + 3.53T + 32T^{2} \) |
| 3 | \( 1 - 13.5T + 243T^{2} \) |
| 11 | \( 1 + 691.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 502.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 991.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 661.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.41e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.75e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.92e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 627.T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.62e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.72e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 4.29e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.59e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 8.90e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.89e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.25e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.89e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.01e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 9.69e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.07e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.24e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.04e5T + 8.58e9T^{2} \) |
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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
−14.94808582298613532937978421080, −13.69119822940869710538499628832, −12.78101222722675327510565152481, −10.83543312987410794957736278271, −9.557668410900609873242043847864, −8.457254639796722827613336821948, −7.49083378784061391460195292367, −4.90416902353730511480470986203, −2.86905804352889549176612019967, 0,
2.86905804352889549176612019967, 4.90416902353730511480470986203, 7.49083378784061391460195292367, 8.457254639796722827613336821948, 9.557668410900609873242043847864, 10.83543312987410794957736278271, 12.78101222722675327510565152481, 13.69119822940869710538499628832, 14.94808582298613532937978421080
