| L(s) = 1 | + (−1.20 − 0.743i)2-s + (−0.0390 − 0.446i)3-s + (0.895 + 1.78i)4-s + (0.378 + 2.20i)5-s + (−0.284 + 0.565i)6-s + (−3.33 − 0.892i)7-s + (0.251 − 2.81i)8-s + (2.75 − 0.486i)9-s + (1.18 − 2.93i)10-s + (0.386 + 0.223i)11-s + (0.762 − 0.469i)12-s + (−0.285 + 3.25i)13-s + (3.34 + 3.55i)14-s + (0.968 − 0.255i)15-s + (−2.39 + 3.20i)16-s + (−4.32 + 3.03i)17-s + ⋯ |
| L(s) = 1 | + (−0.850 − 0.525i)2-s + (−0.0225 − 0.257i)3-s + (0.447 + 0.894i)4-s + (0.169 + 0.985i)5-s + (−0.116 + 0.230i)6-s + (−1.25 − 0.337i)7-s + (0.0890 − 0.996i)8-s + (0.919 − 0.162i)9-s + (0.373 − 0.927i)10-s + (0.116 + 0.0672i)11-s + (0.220 − 0.135i)12-s + (−0.0790 + 0.903i)13-s + (0.894 + 0.948i)14-s + (0.249 − 0.0658i)15-s + (−0.599 + 0.800i)16-s + (−1.04 + 0.734i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.601 - 0.798i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.601 - 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.663228 + 0.330817i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.663228 + 0.330817i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.20 + 0.743i)T \) |
| 5 | \( 1 + (-0.378 - 2.20i)T \) |
| 19 | \( 1 + (-3.60 - 2.44i)T \) |
| good | 3 | \( 1 + (0.0390 + 0.446i)T + (-2.95 + 0.520i)T^{2} \) |
| 7 | \( 1 + (3.33 + 0.892i)T + (6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (-0.386 - 0.223i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.285 - 3.25i)T + (-12.8 - 2.25i)T^{2} \) |
| 17 | \( 1 + (4.32 - 3.03i)T + (5.81 - 15.9i)T^{2} \) |
| 23 | \( 1 + (-3.54 - 7.59i)T + (-14.7 + 17.6i)T^{2} \) |
| 29 | \( 1 + (-3.46 + 0.611i)T + (27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (3.72 - 2.15i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.30 - 5.30i)T + 37iT^{2} \) |
| 41 | \( 1 + (-1.12 - 0.944i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (7.54 + 3.51i)T + (27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (-7.00 - 4.90i)T + (16.0 + 44.1i)T^{2} \) |
| 53 | \( 1 + (-11.6 + 5.41i)T + (34.0 - 40.6i)T^{2} \) |
| 59 | \( 1 + (0.568 - 3.22i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (12.3 + 4.48i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (8.66 + 6.07i)T + (22.9 + 62.9i)T^{2} \) |
| 71 | \( 1 + (-2.32 - 6.39i)T + (-54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-4.66 + 0.407i)T + (71.8 - 12.6i)T^{2} \) |
| 79 | \( 1 + (2.65 + 2.22i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-1.55 + 5.78i)T + (-71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + (6.19 + 7.38i)T + (-15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-7.45 + 5.22i)T + (33.1 - 91.1i)T^{2} \) |
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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
−11.35882160318813792619530981313, −10.38241969095305104626090891652, −9.770772162866403662630178634481, −9.081894701039360808871163776524, −7.55571823472349919082865904487, −6.92559741510865754299730140018, −6.28521864693157522995638803028, −4.02360567548511244222981097226, −3.13156174827735279629148447746, −1.66633110161626396013365647497,
0.67201976752974139367917564415, 2.62042189766238007996800953879, 4.52421468507084529705485834780, 5.51224250582275334688237368796, 6.59318527895630133293224700000, 7.45997066525893658153950596655, 8.748076921647853143391167052263, 9.279637606500833389814029107687, 10.01326751002361098205355757402, 10.87571505333333747262986036786
