| L(s) = 1 | − 7.12·2-s − 9·3-s + 18.7·4-s + 95.7·5-s + 64.0·6-s + 177.·7-s + 94.5·8-s + 81·9-s − 681.·10-s − 146.·11-s − 168.·12-s − 318.·13-s − 1.26e3·14-s − 861.·15-s − 1.27e3·16-s + 713.·17-s − 576.·18-s − 2.46e3·19-s + 1.79e3·20-s − 1.59e3·21-s + 1.04e3·22-s − 4.99e3·23-s − 851.·24-s + 6.03e3·25-s + 2.27e3·26-s − 729·27-s + 3.32e3·28-s + ⋯ |
| L(s) = 1 | − 1.25·2-s − 0.577·3-s + 0.585·4-s + 1.71·5-s + 0.726·6-s + 1.36·7-s + 0.522·8-s + 0.333·9-s − 2.15·10-s − 0.366·11-s − 0.337·12-s − 0.523·13-s − 1.72·14-s − 0.988·15-s − 1.24·16-s + 0.598·17-s − 0.419·18-s − 1.56·19-s + 1.00·20-s − 0.790·21-s + 0.461·22-s − 1.96·23-s − 0.301·24-s + 1.93·25-s + 0.658·26-s − 0.192·27-s + 0.801·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + 9T \) |
| 103 | \( 1 + 1.06e4T \) |
| good | 2 | \( 1 + 7.12T + 32T^{2} \) |
| 5 | \( 1 - 95.7T + 3.12e3T^{2} \) |
| 7 | \( 1 - 177.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 146.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 318.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 713.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.46e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.99e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.00e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.73e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.41e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 2.31e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 6.01e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.08e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.93e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 6.88e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.57e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.16e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.19e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.65e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.71e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.95e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.44e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.76e4T + 8.58e9T^{2} \) |
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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
−10.35274807184007281663916077727, −9.589545368140137642525336589234, −8.560399998521169907918200511869, −7.76619100993030581185865945347, −6.51840222253329764768182441924, −5.47840424102433185481919588779, −4.59434173964869346293856512206, −2.01995603547280887134722086211, −1.61514221274238708065047270951, 0,
1.61514221274238708065047270951, 2.01995603547280887134722086211, 4.59434173964869346293856512206, 5.47840424102433185481919588779, 6.51840222253329764768182441924, 7.76619100993030581185865945347, 8.560399998521169907918200511869, 9.589545368140137642525336589234, 10.35274807184007281663916077727
