| L(s) = 1 | + (1.00 + 0.998i)2-s + (−0.281 + 1.70i)3-s + (0.00793 + 1.99i)4-s + (2.06 − 2.46i)5-s + (−1.98 + 1.43i)6-s + (3.77 + 1.37i)7-s + (−1.98 + 2.01i)8-s + (−2.84 − 0.963i)9-s + (4.53 − 0.405i)10-s + (−1.90 − 2.26i)11-s + (−3.42 − 0.550i)12-s + (−5.98 − 1.05i)13-s + (2.40 + 5.14i)14-s + (3.63 + 4.23i)15-s + (−3.99 + 0.0317i)16-s + (0.333 − 0.576i)17-s + ⋯ |
| L(s) = 1 | + (0.708 + 0.705i)2-s + (−0.162 + 0.986i)3-s + (0.00396 + 0.999i)4-s + (0.925 − 1.10i)5-s + (−0.811 + 0.584i)6-s + (1.42 + 0.518i)7-s + (−0.702 + 0.711i)8-s + (−0.947 − 0.321i)9-s + (1.43 − 0.128i)10-s + (−0.573 − 0.683i)11-s + (−0.987 − 0.158i)12-s + (−1.65 − 0.292i)13-s + (0.644 + 1.37i)14-s + (0.937 + 1.09i)15-s + (−0.999 + 0.00793i)16-s + (0.0807 − 0.139i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0100 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0100 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.30289 + 1.31599i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.30289 + 1.31599i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.00 - 0.998i)T \) |
| 3 | \( 1 + (0.281 - 1.70i)T \) |
| good | 5 | \( 1 + (-2.06 + 2.46i)T + (-0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (-3.77 - 1.37i)T + (5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (1.90 + 2.26i)T + (-1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (5.98 + 1.05i)T + (12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-0.333 + 0.576i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.785 + 0.453i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.52 + 0.917i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-2.05 + 0.362i)T + (27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (6.54 - 2.38i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (-4.94 - 2.85i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.999 + 5.66i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-1.91 - 2.28i)T + (-7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-5.49 - 2.00i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + 2.21iT - 53T^{2} \) |
| 59 | \( 1 + (5.51 - 6.56i)T + (-10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (1.47 - 4.04i)T + (-46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (5.42 + 0.956i)T + (62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-0.0541 + 0.0937i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (6.43 + 11.1i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.85 - 16.1i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-6.91 + 1.22i)T + (77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (6.44 + 11.1i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (5.56 - 4.66i)T + (16.8 - 95.5i)T^{2} \) |
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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
−12.58098124868336639615492355987, −11.80955334854509524998430289819, −10.70633816712499707659132230925, −9.341949177164367556423609342109, −8.660409286178057576077329539249, −7.65806126033467921341166635710, −5.74109574276250573405406300869, −5.17434340096522788381711409132, −4.60137918150966419398421384490, −2.61466753563753878594827740834,
1.81899593655291541599096107636, 2.63542454989495666469197556593, 4.72755915754268148611490728210, 5.67691461018592873366188978329, 6.97907081510096593673516450380, 7.63631954636210137980243310682, 9.500807254697562903354165718678, 10.52093066007562026065327920821, 11.16296481598559997744883247946, 12.10905221296563486435496850920
