L(s) = 1 | + (0.960 + 1.66i)2-s + (−0.760 + 1.31i)3-s + (−0.843 + 1.46i)4-s + (1.00 + 1.73i)5-s − 2.92·6-s + 0.600·8-s + (0.343 + 0.595i)9-s + (−1.92 + 3.33i)10-s + (0.322 − 0.559i)11-s + (−1.28 − 2.22i)12-s − 0.232·13-s − 3.05·15-s + (2.26 + 3.92i)16-s + (−2.85 + 4.95i)17-s + (−0.659 + 1.14i)18-s + (0.5 + 0.866i)19-s + ⋯ |
L(s) = 1 | + (0.678 + 1.17i)2-s + (−0.439 + 0.760i)3-s + (−0.421 + 0.730i)4-s + (0.448 + 0.777i)5-s − 1.19·6-s + 0.212·8-s + (0.114 + 0.198i)9-s + (−0.609 + 1.05i)10-s + (0.0973 − 0.168i)11-s + (−0.370 − 0.641i)12-s − 0.0646·13-s − 0.788·15-s + (0.565 + 0.980i)16-s + (−0.693 + 1.20i)17-s + (−0.155 + 0.269i)18-s + (0.114 + 0.198i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.135461 - 2.13458i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.135461 - 2.13458i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 19 | \( 1 + (-0.5 - 0.866i)T \) |
good | 2 | \( 1 + (-0.960 - 1.66i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (0.760 - 1.31i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-1.00 - 1.73i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.322 + 0.559i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 0.232T + 13T^{2} \) |
| 17 | \( 1 + (2.85 - 4.95i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (2.10 + 3.65i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 3.65T + 29T^{2} \) |
| 31 | \( 1 + (0.876 - 1.51i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.53 + 2.65i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 6.88T + 41T^{2} \) |
| 43 | \( 1 - 3.13T + 43T^{2} \) |
| 47 | \( 1 + (6.25 + 10.8i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.359 + 0.622i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.89 + 3.27i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.56 + 2.70i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.53 - 2.66i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 6.14T + 71T^{2} \) |
| 73 | \( 1 + (-1.97 + 3.42i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.09 - 12.2i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 15.6T + 83T^{2} \) |
| 89 | \( 1 + (4.20 + 7.28i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 15.7T + 97T^{2} \) |
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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
−10.61249942781138984395046591078, −9.860795974289352288665713010392, −8.619354374799653728458595187030, −7.77640629420002480760226587821, −6.72453544557972755676623114859, −6.22470396836390141783452416324, −5.38975353443004709474302072948, −4.51390037062995684297179648117, −3.72943582373696263673101064778, −2.11731539962784838081464039862,
0.891693462403392747688299191443, 1.84209209536421759295749433661, 2.97795195674005942234055810990, 4.26294903820690378678658784840, 5.01654708955312307002277234479, 5.96133628439532686686039874387, 7.02099815605019899051994341863, 7.81448696800159562204111785634, 9.224856810190252711466734678946, 9.626880459578857335698158363968