| L(s) = 1 | + (0.482 + 0.0763i)2-s + (0.517 − 1.01i)3-s + (−1.67 − 0.544i)4-s + (2.05 + 0.885i)5-s + (0.327 − 0.450i)6-s + (−2.59 + 1.32i)7-s + (−1.63 − 0.833i)8-s + (1.00 + 1.37i)9-s + (0.922 + 0.583i)10-s + (−3.15 + 1.00i)11-s + (−1.41 + 1.41i)12-s + (0.457 − 2.89i)13-s + (−1.35 + 0.440i)14-s + (1.96 − 1.62i)15-s + (2.12 + 1.54i)16-s + (−0.824 − 5.20i)17-s + ⋯ |
| L(s) = 1 | + (0.341 + 0.0540i)2-s + (0.298 − 0.586i)3-s + (−0.837 − 0.272i)4-s + (0.918 + 0.395i)5-s + (0.133 − 0.183i)6-s + (−0.982 + 0.500i)7-s + (−0.578 − 0.294i)8-s + (0.333 + 0.458i)9-s + (0.291 + 0.184i)10-s + (−0.952 + 0.303i)11-s + (−0.409 + 0.409i)12-s + (0.126 − 0.801i)13-s + (−0.362 + 0.117i)14-s + (0.506 − 0.419i)15-s + (0.531 + 0.385i)16-s + (−0.199 − 1.26i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.943561 - 0.127304i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.943561 - 0.127304i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-2.05 - 0.885i)T \) |
| 11 | \( 1 + (3.15 - 1.00i)T \) |
| good | 2 | \( 1 + (-0.482 - 0.0763i)T + (1.90 + 0.618i)T^{2} \) |
| 3 | \( 1 + (-0.517 + 1.01i)T + (-1.76 - 2.42i)T^{2} \) |
| 7 | \( 1 + (2.59 - 1.32i)T + (4.11 - 5.66i)T^{2} \) |
| 13 | \( 1 + (-0.457 + 2.89i)T + (-12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (0.824 + 5.20i)T + (-16.1 + 5.25i)T^{2} \) |
| 19 | \( 1 + (-1.26 - 3.89i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (2.12 + 2.12i)T + 23iT^{2} \) |
| 29 | \( 1 + (-0.817 + 2.51i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-5.45 + 3.96i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (0.528 + 1.03i)T + (-21.7 + 29.9i)T^{2} \) |
| 41 | \( 1 + (3.38 - 1.09i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (5.07 - 5.07i)T - 43iT^{2} \) |
| 47 | \( 1 + (3.28 + 1.67i)T + (27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (1.45 + 0.231i)T + (50.4 + 16.3i)T^{2} \) |
| 59 | \( 1 + (1.52 + 0.496i)T + (47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (4.07 - 5.61i)T + (-18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-1.31 + 1.31i)T - 67iT^{2} \) |
| 71 | \( 1 + (2.32 + 1.68i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-6.71 - 13.1i)T + (-42.9 + 59.0i)T^{2} \) |
| 79 | \( 1 + (-12.3 + 8.96i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-6.60 + 1.04i)T + (78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 - 11.1iT - 89T^{2} \) |
| 97 | \( 1 + (1.57 - 9.94i)T + (-92.2 - 29.9i)T^{2} \) |
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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
−15.08947432964851359625323759225, −13.77918640796303740341433700824, −13.30083393921982919948813816282, −12.42718660456437479315115331110, −10.26041334954494817365405518620, −9.596119758788632300337902905423, −8.014639823880894338037795870665, −6.37519687012817315366385278708, −5.16266155503897617673979608367, −2.77622042498335563849113575580,
3.43743930265629117849205769261, 4.84712243607741047695818260564, 6.44740671965072752233149233294, 8.577200900104292030034325145182, 9.529953018976465068372058836770, 10.35766819322416669735252543367, 12.42367489652079561506831703774, 13.30665380659495362606655850106, 13.94760005272241266290047811399, 15.35800334604293694279289754063
