L(s) = 1 | + (−0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + 1.56·5-s + (0.499 − 0.866i)6-s + (−0.5 + 0.866i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (−0.780 − 1.35i)10-s + (1.28 + 2.21i)11-s − 0.999·12-s + (−0.5 + 3.57i)13-s + 0.999·14-s + (0.780 + 1.35i)15-s + (−0.5 − 0.866i)16-s + (−0.0615 + 0.106i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + 0.698·5-s + (0.204 − 0.353i)6-s + (−0.188 + 0.327i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.246 − 0.427i)10-s + (0.386 + 0.668i)11-s − 0.288·12-s + (−0.138 + 0.990i)13-s + 0.267·14-s + (0.201 + 0.349i)15-s + (−0.125 − 0.216i)16-s + (−0.0149 + 0.0258i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 - 0.607i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.794 - 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.31314 + 0.444425i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.31314 + 0.444425i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 + (0.5 - 3.57i)T \) |
good | 5 | \( 1 - 1.56T + 5T^{2} \) |
| 11 | \( 1 + (-1.28 - 2.21i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (0.0615 - 0.106i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.28 + 2.21i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.561 - 0.972i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.06 - 5.30i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + (-1.21 - 2.11i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.62 - 9.73i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-4 + 6.92i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 0.315T + 47T^{2} \) |
| 53 | \( 1 - 7T + 53T^{2} \) |
| 59 | \( 1 + (2.56 - 4.43i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.62 + 9.73i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.56 + 7.90i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-5.68 + 9.84i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 2.43T + 73T^{2} \) |
| 79 | \( 1 + 6.56T + 79T^{2} \) |
| 83 | \( 1 + 5.12T + 83T^{2} \) |
| 89 | \( 1 + (1.71 + 2.97i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.438 + 0.759i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.77555777728642563005537236036, −9.781545111354385708766116907865, −9.394207941828980505377911921659, −8.635503996056534716796047443459, −7.39130873804695683583169536346, −6.36350501819475875124034433767, −5.07914427365449878916410289222, −4.10935545292385730249265189068, −2.81584964406598927557083718058, −1.73122446097542503798652451887,
0.939578901494111557970680690150, 2.55792539260442162263368390087, 3.98302415572056521904535188870, 5.57700340090638447950390809506, 6.10258884181386150685882078403, 7.18662181021587355110342057432, 7.986876294054291129943389105104, 8.830066507560476611301431837426, 9.753355012788214259759840078212, 10.42384441318494637947263695798