| L(s) = 1 | + (3.97 − 6.88i)5-s − 5.83·7-s − 9.98·11-s + (21.1 − 12.2i)13-s + (−1.10 + 1.92i)17-s + (18.6 + 3.41i)19-s + (9.98 + 17.2i)23-s + (−19.0 − 33.0i)25-s + (27.4 − 15.8i)29-s + 5.42i·31-s + (−23.1 + 40.1i)35-s − 62.6i·37-s + (−30.8 − 17.8i)41-s + (8.03 − 13.9i)43-s + (−28.4 − 49.3i)47-s + ⋯ |
| L(s) = 1 | + (0.794 − 1.37i)5-s − 0.833·7-s − 0.907·11-s + (1.62 − 0.940i)13-s + (−0.0652 + 0.112i)17-s + (0.983 + 0.179i)19-s + (0.434 + 0.752i)23-s + (−0.762 − 1.32i)25-s + (0.945 − 0.545i)29-s + 0.174i·31-s + (−0.662 + 1.14i)35-s − 1.69i·37-s + (−0.752 − 0.434i)41-s + (0.186 − 0.323i)43-s + (−0.605 − 1.04i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.570 + 0.821i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.570 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.775160027\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.775160027\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-18.6 - 3.41i)T \) |
| good | 5 | \( 1 + (-3.97 + 6.88i)T + (-12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + 5.83T + 49T^{2} \) |
| 11 | \( 1 + 9.98T + 121T^{2} \) |
| 13 | \( 1 + (-21.1 + 12.2i)T + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + (1.10 - 1.92i)T + (-144.5 - 250. i)T^{2} \) |
| 23 | \( 1 + (-9.98 - 17.2i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-27.4 + 15.8i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 - 5.42iT - 961T^{2} \) |
| 37 | \( 1 + 62.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (30.8 + 17.8i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-8.03 + 13.9i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (28.4 + 49.3i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-12.8 + 7.40i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (80.2 + 46.3i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-27.5 - 47.6i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-47.4 + 27.3i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + (93.3 + 53.8i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (56.0 - 97.1i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-33.4 - 19.3i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + 17.7T + 6.88e3T^{2} \) |
| 89 | \( 1 + (118. - 68.5i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (29.0 + 16.7i)T + (4.70e3 + 8.14e3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.071669568818195600454807292299, −8.479097886687431931871842744136, −7.67639770065711441376435152186, −6.45205796453430743659726405651, −5.56028940668273379799467115409, −5.26324718299874416254153965395, −3.89125767080292713082060920965, −2.94203271370878338536203107766, −1.52845967254116667787341662976, −0.51789154685792974679993236720,
1.44431274537008275982485021528, 2.88355436510924681136529007308, 3.17996716704298891263474472484, 4.61189390536141201499747195786, 5.82384887163664201443393189187, 6.50327954007519917762484396267, 6.90016175999565020185434525478, 8.058998174355286638722770272541, 9.010369772700265022594754278504, 9.820633436260635655291573847232
