| L(s) = 1 | + (−0.0211 − 1.41i)2-s + (0.382 + 1.68i)3-s + (−1.99 + 0.0598i)4-s + (2.03 − 0.618i)5-s + (2.38 − 0.577i)6-s + (0.647 − 3.25i)7-s + (0.126 + 2.82i)8-s + (−2.70 + 1.29i)9-s + (−0.918 − 2.87i)10-s + (0.158 − 1.61i)11-s + (−0.866 − 3.35i)12-s + (3.48 + 1.05i)13-s + (−4.61 − 0.847i)14-s + (1.82 + 3.20i)15-s + (3.99 − 0.239i)16-s + (1.96 − 0.813i)17-s + ⋯ |
| L(s) = 1 | + (−0.0149 − 0.999i)2-s + (0.221 + 0.975i)3-s + (−0.999 + 0.0299i)4-s + (0.912 − 0.276i)5-s + (0.971 − 0.235i)6-s + (0.244 − 1.23i)7-s + (0.0448 + 0.998i)8-s + (−0.902 + 0.431i)9-s + (−0.290 − 0.908i)10-s + (0.0478 − 0.485i)11-s + (−0.250 − 0.968i)12-s + (0.967 + 0.293i)13-s + (−1.23 − 0.226i)14-s + (0.471 + 0.828i)15-s + (0.998 − 0.0597i)16-s + (0.476 − 0.197i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.542 + 0.840i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.542 + 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.36720 - 0.744898i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.36720 - 0.744898i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.0211 + 1.41i)T \) |
| 3 | \( 1 + (-0.382 - 1.68i)T \) |
| good | 5 | \( 1 + (-2.03 + 0.618i)T + (4.15 - 2.77i)T^{2} \) |
| 7 | \( 1 + (-0.647 + 3.25i)T + (-6.46 - 2.67i)T^{2} \) |
| 11 | \( 1 + (-0.158 + 1.61i)T + (-10.7 - 2.14i)T^{2} \) |
| 13 | \( 1 + (-3.48 - 1.05i)T + (10.8 + 7.22i)T^{2} \) |
| 17 | \( 1 + (-1.96 + 0.813i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (-3.61 + 1.93i)T + (10.5 - 15.7i)T^{2} \) |
| 23 | \( 1 + (1.30 - 1.95i)T + (-8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (-0.175 - 1.77i)T + (-28.4 + 5.65i)T^{2} \) |
| 31 | \( 1 + (-7.41 + 7.41i)T - 31iT^{2} \) |
| 37 | \( 1 + (-4.45 - 2.38i)T + (20.5 + 30.7i)T^{2} \) |
| 41 | \( 1 + (6.01 - 9.00i)T + (-15.6 - 37.8i)T^{2} \) |
| 43 | \( 1 + (5.84 + 7.12i)T + (-8.38 + 42.1i)T^{2} \) |
| 47 | \( 1 + (7.55 - 3.12i)T + (33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (-0.0669 + 0.679i)T + (-51.9 - 10.3i)T^{2} \) |
| 59 | \( 1 + (6.68 - 2.02i)T + (49.0 - 32.7i)T^{2} \) |
| 61 | \( 1 + (5.91 - 7.21i)T + (-11.9 - 59.8i)T^{2} \) |
| 67 | \( 1 + (0.592 + 0.486i)T + (13.0 + 65.7i)T^{2} \) |
| 71 | \( 1 + (1.70 - 8.56i)T + (-65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (-0.735 + 0.146i)T + (67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (6.16 - 14.8i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (0.488 - 0.260i)T + (46.1 - 69.0i)T^{2} \) |
| 89 | \( 1 + (-9.97 + 6.66i)T + (34.0 - 82.2i)T^{2} \) |
| 97 | \( 1 + (2.86 + 2.86i)T + 97iT^{2} \) |
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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
−11.19010302549774752140748680015, −10.05391778176233296508487155102, −9.810524480174469041666651799175, −8.766879247394191439378671621983, −7.86148065809986962576319915001, −6.04190189305157321706322455827, −4.98465024872901574381362583733, −4.02995729363201704428584780750, −3.02954548650837829361478325741, −1.28242523951838294295916162364,
1.65874700558221295291735804003, 3.17502244587479699827136918314, 5.13163836416092538124969689577, 6.02185379278205395239673500216, 6.51128850832439198058834644722, 7.81899502332451167377464851917, 8.482374486867790654557843904630, 9.346953799326690522990831481337, 10.29119760577328730408658267420, 11.85559249948534985867531405703
