| L(s) = 1 | + (2.98 − 0.291i)3-s + 4.46i·7-s + (8.82 − 1.74i)9-s − 17.8i·11-s − 11.0i·13-s − 0.794·17-s + 26.5·19-s + (1.30 + 13.3i)21-s − 14.9·23-s + (25.8 − 7.77i)27-s + 5.58i·29-s − 53.1·31-s + (−5.21 − 53.3i)33-s − 51.7i·37-s + (−3.21 − 32.8i)39-s + ⋯ |
| L(s) = 1 | + (0.995 − 0.0972i)3-s + 0.637i·7-s + (0.981 − 0.193i)9-s − 1.62i·11-s − 0.846i·13-s − 0.0467·17-s + 1.39·19-s + (0.0619 + 0.634i)21-s − 0.649·23-s + (0.957 − 0.287i)27-s + 0.192i·29-s − 1.71·31-s + (−0.157 − 1.61i)33-s − 1.39i·37-s + (−0.0823 − 0.842i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.532 + 0.846i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.532 + 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.798826248\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.798826248\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.98 + 0.291i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 4.46iT - 49T^{2} \) |
| 11 | \( 1 + 17.8iT - 121T^{2} \) |
| 13 | \( 1 + 11.0iT - 169T^{2} \) |
| 17 | \( 1 + 0.794T + 289T^{2} \) |
| 19 | \( 1 - 26.5T + 361T^{2} \) |
| 23 | \( 1 + 14.9T + 529T^{2} \) |
| 29 | \( 1 - 5.58iT - 841T^{2} \) |
| 31 | \( 1 + 53.1T + 961T^{2} \) |
| 37 | \( 1 + 51.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 67.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 40.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 12.3T + 2.20e3T^{2} \) |
| 53 | \( 1 - 37.0T + 2.80e3T^{2} \) |
| 59 | \( 1 - 61.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 97.8T + 3.72e3T^{2} \) |
| 67 | \( 1 - 3.02iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 57.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 31.4iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 2.16T + 6.24e3T^{2} \) |
| 83 | \( 1 - 13.0T + 6.88e3T^{2} \) |
| 89 | \( 1 + 173. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 91.6iT - 9.40e3T^{2} \) |
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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
−9.081396784677718374032229264488, −8.794899159630741209108614391002, −7.83313846026074267629214489325, −7.23678158693688035078758091347, −5.87469615255421603304357596028, −5.39375250671364641672048729894, −3.81332566597199942045762026480, −3.20496602478893776997642501097, −2.20085045742550634746972731375, −0.75966842138599012861648214335,
1.41431816804510256343094673549, 2.33788962159521963677395378394, 3.61788524357812434512695590789, 4.32293853968446834633062391757, 5.21263436072394893642782344597, 6.77103686714444705134803400503, 7.25805509817469674594558626087, 7.961181232771998425588600560052, 8.976616205336530375270418933977, 9.845801734127305384782188972440
