| L(s) = 1 | + (−1.61 + 2.19i)2-s + (1.47 − 0.278i)3-s + (−1.59 − 5.17i)4-s + (−2.06 + 0.849i)5-s + (−1.77 + 3.67i)6-s + (1.83 + 1.90i)7-s + (8.78 + 3.07i)8-s + (−0.700 + 0.274i)9-s + (1.48 − 5.90i)10-s + (−1.69 + 4.31i)11-s + (−3.79 − 7.18i)12-s + (−3.39 − 0.382i)13-s + (−7.14 + 0.930i)14-s + (−2.81 + 1.82i)15-s + (−11.9 + 8.17i)16-s + (0.515 + 0.0192i)17-s + ⋯ |
| L(s) = 1 | + (−1.14 + 1.54i)2-s + (0.850 − 0.160i)3-s + (−0.798 − 2.58i)4-s + (−0.925 + 0.379i)5-s + (−0.723 + 1.50i)6-s + (0.692 + 0.721i)7-s + (3.10 + 1.08i)8-s + (−0.233 + 0.0916i)9-s + (0.469 − 1.86i)10-s + (−0.510 + 1.30i)11-s + (−1.09 − 2.07i)12-s + (−0.940 − 0.105i)13-s + (−1.90 + 0.248i)14-s + (−0.725 + 0.471i)15-s + (−2.99 + 2.04i)16-s + (0.124 + 0.00467i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0736i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0736i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0215027 - 0.582776i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0215027 - 0.582776i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (2.06 - 0.849i)T \) |
| 7 | \( 1 + (-1.83 - 1.90i)T \) |
| good | 2 | \( 1 + (1.61 - 2.19i)T + (-0.589 - 1.91i)T^{2} \) |
| 3 | \( 1 + (-1.47 + 0.278i)T + (2.79 - 1.09i)T^{2} \) |
| 11 | \( 1 + (1.69 - 4.31i)T + (-8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (3.39 + 0.382i)T + (12.6 + 2.89i)T^{2} \) |
| 17 | \( 1 + (-0.515 - 0.0192i)T + (16.9 + 1.27i)T^{2} \) |
| 19 | \( 1 + (1.16 - 2.01i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.120 - 3.23i)T + (-22.9 + 1.71i)T^{2} \) |
| 29 | \( 1 + (-0.821 + 0.187i)T + (26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (0.940 - 0.543i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-7.67 + 4.05i)T + (20.8 - 30.5i)T^{2} \) |
| 41 | \( 1 + (0.358 + 0.744i)T + (-25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (1.31 + 3.75i)T + (-33.6 + 26.8i)T^{2} \) |
| 47 | \( 1 + (-9.96 - 7.35i)T + (13.8 + 44.9i)T^{2} \) |
| 53 | \( 1 + (1.60 + 0.849i)T + (29.8 + 43.7i)T^{2} \) |
| 59 | \( 1 + (0.338 + 4.51i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (0.194 - 0.629i)T + (-50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (-10.6 - 2.84i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (3.21 - 14.0i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-0.471 + 0.347i)T + (21.5 - 69.7i)T^{2} \) |
| 79 | \( 1 + (1.16 + 0.670i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1.63 + 14.4i)T + (-80.9 + 18.4i)T^{2} \) |
| 89 | \( 1 + (2.10 + 5.35i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (3.98 + 3.98i)T + 97iT^{2} \) |
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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
−12.68123450521237776570371267668, −11.37146004552891520362012126105, −10.21618787763352605970338848104, −9.282269009808960974575698168000, −8.387053762243573920447052876254, −7.65468341203421914559061477054, −7.26237778836361734657083960469, −5.69435437631972699299266548035, −4.60963457661630379030294611820, −2.23949874785681840362547029001,
0.60985644271094337233455530990, 2.56693506218654586740986904690, 3.56806989726500954228756264249, 4.60850611919666814850292251211, 7.42872125661405872616954890001, 8.210544195894148253705219864098, 8.642463197837116126222679081490, 9.673591917082604178766572177450, 10.78191265630142269216319814110, 11.34453025786945065694523920462
