| L(s) = 1 | + (−0.976 + 0.214i)2-s + (0.745 − 1.56i)3-s + (0.907 − 0.419i)4-s + (−1.05 − 0.291i)5-s + (−0.391 + 1.68i)6-s + (1.62 − 1.53i)7-s + (−0.796 + 0.605i)8-s + (−1.88 − 2.33i)9-s + (1.08 + 0.0590i)10-s + (−0.613 − 0.722i)11-s + (0.0200 − 1.73i)12-s + (−0.373 − 1.10i)13-s + (−1.25 + 1.85i)14-s + (−1.23 + 1.42i)15-s + (0.647 − 0.762i)16-s + (1.44 − 1.52i)17-s + ⋯ |
| L(s) = 1 | + (−0.690 + 0.152i)2-s + (0.430 − 0.902i)3-s + (0.453 − 0.209i)4-s + (−0.469 − 0.130i)5-s + (−0.159 + 0.688i)6-s + (0.613 − 0.581i)7-s + (−0.281 + 0.213i)8-s + (−0.629 − 0.776i)9-s + (0.344 + 0.0186i)10-s + (−0.185 − 0.217i)11-s + (0.00578 − 0.499i)12-s + (−0.103 − 0.307i)13-s + (−0.335 + 0.494i)14-s + (−0.320 + 0.368i)15-s + (0.161 − 0.190i)16-s + (0.351 − 0.370i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.290 + 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.290 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.569880 - 0.768597i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.569880 - 0.768597i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.976 - 0.214i)T \) |
| 3 | \( 1 + (-0.745 + 1.56i)T \) |
| 59 | \( 1 + (-3.23 + 6.96i)T \) |
| good | 5 | \( 1 + (1.05 + 0.291i)T + (4.28 + 2.57i)T^{2} \) |
| 7 | \( 1 + (-1.62 + 1.53i)T + (0.378 - 6.98i)T^{2} \) |
| 11 | \( 1 + (0.613 + 0.722i)T + (-1.77 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.373 + 1.10i)T + (-10.3 + 7.86i)T^{2} \) |
| 17 | \( 1 + (-1.44 + 1.52i)T + (-0.920 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-0.0477 - 0.119i)T + (-13.7 + 13.0i)T^{2} \) |
| 23 | \( 1 + (1.83 - 0.199i)T + (22.4 - 4.94i)T^{2} \) |
| 29 | \( 1 + (-0.659 + 2.99i)T + (-26.3 - 12.1i)T^{2} \) |
| 31 | \( 1 + (0.531 + 0.211i)T + (22.5 + 21.3i)T^{2} \) |
| 37 | \( 1 + (-2.73 + 3.59i)T + (-9.89 - 35.6i)T^{2} \) |
| 41 | \( 1 + (-0.194 + 1.79i)T + (-40.0 - 8.81i)T^{2} \) |
| 43 | \( 1 + (-5.30 - 4.50i)T + (6.95 + 42.4i)T^{2} \) |
| 47 | \( 1 + (1.66 + 5.99i)T + (-40.2 + 24.2i)T^{2} \) |
| 53 | \( 1 + (-8.59 + 0.466i)T + (52.6 - 5.73i)T^{2} \) |
| 61 | \( 1 + (-0.184 - 0.840i)T + (-55.3 + 25.6i)T^{2} \) |
| 67 | \( 1 + (-4.68 - 6.16i)T + (-17.9 + 64.5i)T^{2} \) |
| 71 | \( 1 + (7.97 - 2.21i)T + (60.8 - 36.6i)T^{2} \) |
| 73 | \( 1 + (-12.2 - 8.30i)T + (27.0 + 67.8i)T^{2} \) |
| 79 | \( 1 + (-0.607 - 3.70i)T + (-74.8 + 25.2i)T^{2} \) |
| 83 | \( 1 + (-3.75 - 7.07i)T + (-46.5 + 68.6i)T^{2} \) |
| 89 | \( 1 + (-13.2 - 2.91i)T + (80.7 + 37.3i)T^{2} \) |
| 97 | \( 1 + (5.27 - 3.57i)T + (35.9 - 90.1i)T^{2} \) |
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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
−11.29503885980310832158600633832, −10.20640086293317581640487466098, −9.150305295427660252774054776097, −8.020736584137542669269489993394, −7.78341104791511312855631281121, −6.72761088556720952893073715552, −5.53500259753829167686634021169, −3.89631741750453796099621807982, −2.36433771461735273386687524490, −0.794464992741667602089934767619,
2.12053855871893400940042609995, 3.45375844797070079406197747849, 4.65081859524664386438807941220, 5.84554127774639553314700930023, 7.41697235719567494036076419482, 8.203288323682848269690127342656, 8.976485590695720454017430633304, 9.857456168048328774001585085763, 10.71914408455304654651966111439, 11.51192763833224280555559482390
