| L(s) = 1 | + (−0.571 + 1.31i)2-s + (1.68 + 0.416i)3-s + (−0.0295 − 0.0318i)4-s + (2.61 + 0.394i)5-s + (−1.50 + 1.96i)6-s + (−1.32 − 1.22i)7-s + (−2.63 + 0.923i)8-s + (2.65 + 1.39i)9-s + (−2.01 + 3.20i)10-s + (−0.291 + 0.250i)11-s + (−0.0364 − 0.0658i)12-s + (1.83 − 2.69i)13-s + (2.36 − 1.03i)14-s + (4.23 + 1.75i)15-s + (0.305 − 4.07i)16-s + (0.359 + 0.359i)17-s + ⋯ |
| L(s) = 1 | + (−0.404 + 0.926i)2-s + (0.970 + 0.240i)3-s + (−0.0147 − 0.0159i)4-s + (1.16 + 0.176i)5-s + (−0.615 + 0.802i)6-s + (−0.500 − 0.464i)7-s + (−0.933 + 0.326i)8-s + (0.884 + 0.466i)9-s + (−0.635 + 1.01i)10-s + (−0.0878 + 0.0756i)11-s + (−0.0105 − 0.0190i)12-s + (0.508 − 0.746i)13-s + (0.632 − 0.275i)14-s + (1.09 + 0.452i)15-s + (0.0763 − 1.01i)16-s + (0.0871 + 0.0871i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.000120 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.000120 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.13799 + 1.13813i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.13799 + 1.13813i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.68 - 0.416i)T \) |
| 29 | \( 1 + (-2.80 - 4.59i)T \) |
| good | 2 | \( 1 + (0.571 - 1.31i)T + (-1.36 - 1.46i)T^{2} \) |
| 5 | \( 1 + (-2.61 - 0.394i)T + (4.77 + 1.47i)T^{2} \) |
| 7 | \( 1 + (1.32 + 1.22i)T + (0.523 + 6.98i)T^{2} \) |
| 11 | \( 1 + (0.291 - 0.250i)T + (1.63 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-1.83 + 2.69i)T + (-4.74 - 12.1i)T^{2} \) |
| 17 | \( 1 + (-0.359 - 0.359i)T + 17iT^{2} \) |
| 19 | \( 1 + (6.57 + 4.13i)T + (8.24 + 17.1i)T^{2} \) |
| 23 | \( 1 + (-0.0597 - 0.0234i)T + (16.8 + 15.6i)T^{2} \) |
| 31 | \( 1 + (7.73 - 5.70i)T + (9.13 - 29.6i)T^{2} \) |
| 37 | \( 1 + (1.91 + 5.46i)T + (-28.9 + 23.0i)T^{2} \) |
| 41 | \( 1 + (1.60 + 0.429i)T + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (-1.21 + 1.64i)T + (-12.6 - 41.0i)T^{2} \) |
| 47 | \( 1 + (0.331 + 0.385i)T + (-7.00 + 46.4i)T^{2} \) |
| 53 | \( 1 + (-1.68 - 1.34i)T + (11.7 + 51.6i)T^{2} \) |
| 59 | \( 1 + (-3.87 + 2.23i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-9.16 + 0.342i)T + (60.8 - 4.55i)T^{2} \) |
| 67 | \( 1 + (-13.5 + 1.01i)T + (66.2 - 9.98i)T^{2} \) |
| 71 | \( 1 + (14.0 - 6.76i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (0.264 - 2.34i)T + (-71.1 - 16.2i)T^{2} \) |
| 79 | \( 1 + (-1.45 + 7.66i)T + (-73.5 - 28.8i)T^{2} \) |
| 83 | \( 1 + (-0.0736 + 0.238i)T + (-68.5 - 46.7i)T^{2} \) |
| 89 | \( 1 + (-8.94 + 1.00i)T + (86.7 - 19.8i)T^{2} \) |
| 97 | \( 1 + (-5.87 - 11.1i)T + (-54.6 + 80.1i)T^{2} \) |
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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
−12.69150343154975398165149336038, −10.81617554502628082766125034764, −10.10713470617680834704918597842, −9.036281946942202240560957712543, −8.491859441436434052076831944776, −7.22851130387191051823861239993, −6.54099731733275448511824853713, −5.33713889710104009326344688462, −3.54418447034826187637853425847, −2.30643741097307029312726568687,
1.70915585298720106321220332223, 2.48324544818915970388784083724, 3.87307465276385202612206610405, 5.90139968771642240680496746865, 6.60362586266134100022513795908, 8.306283714331957255174377281149, 9.172147295436924855578708583308, 9.740672422232677045331104138618, 10.52147589835624136804808317879, 11.76684588637033914799688602016
