L(s) = 1 | + (1.35 − 0.416i)2-s + (−0.244 + 0.590i)3-s + (1.65 − 1.12i)4-s + (−0.220 + 0.0913i)5-s + (−0.0844 + 0.899i)6-s + (0.707 − 0.707i)7-s + (1.76 − 2.21i)8-s + (1.83 + 1.83i)9-s + (−0.259 + 0.215i)10-s + (−0.352 − 0.851i)11-s + (0.260 + 1.25i)12-s + (−1.31 − 0.545i)13-s + (0.660 − 1.25i)14-s − 0.152i·15-s + (1.46 − 3.72i)16-s + 3.60i·17-s + ⋯ |
L(s) = 1 | + (0.955 − 0.294i)2-s + (−0.141 + 0.340i)3-s + (0.826 − 0.563i)4-s + (−0.0986 + 0.0408i)5-s + (−0.0344 + 0.367i)6-s + (0.267 − 0.267i)7-s + (0.623 − 0.781i)8-s + (0.610 + 0.610i)9-s + (−0.0821 + 0.0680i)10-s + (−0.106 − 0.256i)11-s + (0.0752 + 0.361i)12-s + (−0.365 − 0.151i)13-s + (0.176 − 0.334i)14-s − 0.0393i·15-s + (0.365 − 0.930i)16-s + 0.874i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 + 0.268i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.963 + 0.268i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.01176 - 0.274746i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.01176 - 0.274746i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.35 + 0.416i)T \) |
| 7 | \( 1 + (-0.707 + 0.707i)T \) |
good | 3 | \( 1 + (0.244 - 0.590i)T + (-2.12 - 2.12i)T^{2} \) |
| 5 | \( 1 + (0.220 - 0.0913i)T + (3.53 - 3.53i)T^{2} \) |
| 11 | \( 1 + (0.352 + 0.851i)T + (-7.77 + 7.77i)T^{2} \) |
| 13 | \( 1 + (1.31 + 0.545i)T + (9.19 + 9.19i)T^{2} \) |
| 17 | \( 1 - 3.60iT - 17T^{2} \) |
| 19 | \( 1 + (3.74 + 1.55i)T + (13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (0.802 + 0.802i)T + 23iT^{2} \) |
| 29 | \( 1 + (-0.583 + 1.40i)T + (-20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 + 5.77T + 31T^{2} \) |
| 37 | \( 1 + (2.19 - 0.910i)T + (26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (-4.11 - 4.11i)T + 41iT^{2} \) |
| 43 | \( 1 + (2.07 + 5.01i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + 0.582iT - 47T^{2} \) |
| 53 | \( 1 + (-1.75 - 4.23i)T + (-37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (7.70 - 3.19i)T + (41.7 - 41.7i)T^{2} \) |
| 61 | \( 1 + (-4.59 + 11.0i)T + (-43.1 - 43.1i)T^{2} \) |
| 67 | \( 1 + (3.28 - 7.92i)T + (-47.3 - 47.3i)T^{2} \) |
| 71 | \( 1 + (-10.8 + 10.8i)T - 71iT^{2} \) |
| 73 | \( 1 + (-9.19 - 9.19i)T + 73iT^{2} \) |
| 79 | \( 1 - 5.19iT - 79T^{2} \) |
| 83 | \( 1 + (-9.73 - 4.03i)T + (58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (-7.11 + 7.11i)T - 89iT^{2} \) |
| 97 | \( 1 + 13.2T + 97T^{2} \) |
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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
−12.36417639337934564793192241152, −11.12113616659095776415195394727, −10.64974138491144678528213421348, −9.650984841815102216790557545525, −8.041428058190397849579081672509, −7.00124183753048856493982514406, −5.74147636254752211161488733196, −4.67767210462556832887335887388, −3.71740997542298460032190757018, −1.99237847481692191168527346196,
2.12754267525031540590401772199, 3.80253688787973389638418188913, 4.92157451474387310511121395082, 6.12890791451371851088770287969, 7.08167428759787283357379099534, 7.963349491414491729402884347008, 9.338374010993986115330071882629, 10.62033954122241356963336672360, 11.76827995096902210031748365209, 12.33900822016598049338605674513