L(s) = 1 | + (0.909 − 1.08i)2-s + (0.456 − 1.67i)3-s + (−0.347 − 1.96i)4-s + (−1.39 − 2.01i)6-s + (−2.44 − 1.41i)8-s + (−2.58 − 1.52i)9-s + (2.18 + 6.01i)11-s + (−3.44 − 0.317i)12-s + (−3.75 + 1.36i)16-s + (7.09 − 4.09i)17-s + (−4.00 + 1.41i)18-s + (−0.511 + 0.885i)19-s + (8.50 + 3.09i)22-s + (−3.48 + 3.44i)24-s + (−3.83 − 3.21i)25-s + ⋯ |
L(s) = 1 | + (0.642 − 0.766i)2-s + (0.263 − 0.964i)3-s + (−0.173 − 0.984i)4-s + (−0.569 − 0.821i)6-s + (−0.866 − 0.500i)8-s + (−0.861 − 0.507i)9-s + (0.659 + 1.81i)11-s + (−0.995 − 0.0917i)12-s + (−0.939 + 0.342i)16-s + (1.72 − 0.993i)17-s + (−0.942 + 0.333i)18-s + (−0.117 + 0.203i)19-s + (1.81 + 0.659i)22-s + (−0.710 + 0.703i)24-s + (−0.766 − 0.642i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.486 + 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.486 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.870576 - 1.48168i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.870576 - 1.48168i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.909 + 1.08i)T \) |
| 3 | \( 1 + (-0.456 + 1.67i)T \) |
good | 5 | \( 1 + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-2.18 - 6.01i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-7.09 + 4.09i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.511 - 0.885i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-8.15 - 9.71i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-2.07 + 0.754i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + (3.84 - 10.5i)T + (-45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (4.53 - 3.80i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.68 + 8.12i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (10.9 - 13.0i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (11.0 + 6.38i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (14.0 - 5.12i)T + (74.3 - 62.3i)T^{2} \) |
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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
−12.20965167024924706109042214358, −11.48590301834963993813959406614, −9.976520441584837896150176198718, −9.391901780765024804289587528027, −7.81977568006949544286182679416, −6.83955656162311161219205237048, −5.67053389201541085779608285464, −4.30848959087234503391354529459, −2.81530497815431828271566832411, −1.47049972213918293703879680815,
3.23277142599543374205074813262, 3.94602436640490176007187748870, 5.47671829736626169996222406899, 6.08861985760862356122981438412, 7.77622878214274477604102826090, 8.564019066307208717957904922872, 9.486342300852992227792493851666, 10.83198692044434170845407402044, 11.66394992909315630016896658656, 12.81664473785768302921280385514