| L(s) = 1 | + (−0.511 − 0.108i)2-s + (1.18 + 1.26i)3-s + (−1.57 − 0.702i)4-s + (1.99 + 1.00i)5-s + (−0.468 − 0.775i)6-s + (−1.68 + 2.92i)7-s + (1.57 + 1.14i)8-s + (−0.192 + 2.99i)9-s + (−0.913 − 0.730i)10-s + (−4.04 − 0.859i)11-s + (−0.981 − 2.82i)12-s + (4.96 − 1.05i)13-s + (1.18 − 1.31i)14-s + (1.10 + 3.71i)15-s + (1.62 + 1.80i)16-s + (0.0810 + 0.0589i)17-s + ⋯ |
| L(s) = 1 | + (−0.361 − 0.0768i)2-s + (0.684 + 0.729i)3-s + (−0.788 − 0.351i)4-s + (0.893 + 0.448i)5-s + (−0.191 − 0.316i)6-s + (−0.638 + 1.10i)7-s + (0.557 + 0.405i)8-s + (−0.0640 + 0.997i)9-s + (−0.288 − 0.230i)10-s + (−1.21 − 0.259i)11-s + (−0.283 − 0.815i)12-s + (1.37 − 0.292i)13-s + (0.315 − 0.350i)14-s + (0.284 + 0.958i)15-s + (0.407 + 0.452i)16-s + (0.0196 + 0.0142i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.264 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.264 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.887874 + 0.676878i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.887874 + 0.676878i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.18 - 1.26i)T \) |
| 5 | \( 1 + (-1.99 - 1.00i)T \) |
| good | 2 | \( 1 + (0.511 + 0.108i)T + (1.82 + 0.813i)T^{2} \) |
| 7 | \( 1 + (1.68 - 2.92i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (4.04 + 0.859i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (-4.96 + 1.05i)T + (11.8 - 5.28i)T^{2} \) |
| 17 | \( 1 + (-0.0810 - 0.0589i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-2.55 - 1.85i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (1.31 - 1.46i)T + (-2.40 - 22.8i)T^{2} \) |
| 29 | \( 1 + (0.709 + 6.74i)T + (-28.3 + 6.02i)T^{2} \) |
| 31 | \( 1 + (0.254 - 2.42i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (-0.930 + 2.86i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-5.68 + 1.20i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (-5.61 + 9.73i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.973 + 9.26i)T + (-45.9 + 9.77i)T^{2} \) |
| 53 | \( 1 + (-1.84 + 1.34i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (8.28 - 1.76i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (3.39 + 0.722i)T + (55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (-1.41 + 13.4i)T + (-65.5 - 13.9i)T^{2} \) |
| 71 | \( 1 + (-6.75 + 4.90i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-3.28 - 10.1i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-1.50 - 14.2i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (5.66 - 2.52i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (2.14 + 6.60i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-0.182 - 1.73i)T + (-94.8 + 20.1i)T^{2} \) |
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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
−12.75534977481930577200072539318, −10.98367157569573364480250492791, −10.27682345279500931357838056018, −9.500120879841051521762778208368, −8.810730796566997642237529123634, −7.905840282988532224081394298631, −5.88038586019266650063871526577, −5.36865979086023485207798165063, −3.59886572051882784977769644820, −2.31772261807104720033054383638,
1.09542682420657432822763358015, 3.09215279227082288418555892057, 4.45407199906233043760855243911, 6.05087064054563987134827158302, 7.25629511136241971012090115540, 8.122841183628747148983115397488, 9.101074625750853191104190603445, 9.790359317621453194429208892377, 10.81695220015392544435919331197, 12.65492119482943955220066618291
