L(s) = 1 | + (2.05 + 1.49i)2-s + (−1.94 + 1.41i)3-s + (1.37 + 4.24i)4-s + 1.44·5-s − 6.10·6-s + (0.616 + 1.89i)7-s + (−1.93 + 5.95i)8-s + (0.854 − 2.62i)9-s + (2.97 + 2.15i)10-s + (−0.941 − 2.89i)11-s + (−8.66 − 6.29i)12-s + (0.809 − 0.587i)13-s + (−1.56 + 4.82i)14-s + (−2.80 + 2.03i)15-s + (−5.65 + 4.10i)16-s + (0.846 − 2.60i)17-s + ⋯ |
L(s) = 1 | + (1.45 + 1.05i)2-s + (−1.12 + 0.814i)3-s + (0.689 + 2.12i)4-s + 0.646·5-s − 2.49·6-s + (0.233 + 0.717i)7-s + (−0.683 + 2.10i)8-s + (0.284 − 0.876i)9-s + (0.939 + 0.682i)10-s + (−0.283 − 0.874i)11-s + (−2.50 − 1.81i)12-s + (0.224 − 0.163i)13-s + (−0.419 + 1.29i)14-s + (−0.724 + 0.526i)15-s + (−1.41 + 1.02i)16-s + (0.205 − 0.631i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.912 - 0.408i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.912 - 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.477839 + 2.24011i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.477839 + 2.24011i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 + (-0.809 + 0.587i)T \) |
| 31 | \( 1 + (5.11 + 2.19i)T \) |
good | 2 | \( 1 + (-2.05 - 1.49i)T + (0.618 + 1.90i)T^{2} \) |
| 3 | \( 1 + (1.94 - 1.41i)T + (0.927 - 2.85i)T^{2} \) |
| 5 | \( 1 - 1.44T + 5T^{2} \) |
| 7 | \( 1 + (-0.616 - 1.89i)T + (-5.66 + 4.11i)T^{2} \) |
| 11 | \( 1 + (0.941 + 2.89i)T + (-8.89 + 6.46i)T^{2} \) |
| 17 | \( 1 + (-0.846 + 2.60i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (0.156 + 0.113i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (0.579 - 1.78i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-6.38 - 4.63i)T + (8.96 + 27.5i)T^{2} \) |
| 37 | \( 1 - 7.58T + 37T^{2} \) |
| 41 | \( 1 + (-6.48 - 4.70i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (4.59 + 3.33i)T + (13.2 + 40.8i)T^{2} \) |
| 47 | \( 1 + (-3.83 + 2.78i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (0.845 - 2.60i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (7.58 - 5.50i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + 2.54T + 61T^{2} \) |
| 67 | \( 1 - 9.61T + 67T^{2} \) |
| 71 | \( 1 + (3.60 - 11.0i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (3.47 + 10.6i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (1.41 - 4.36i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (0.975 + 0.708i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (5.04 + 15.5i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (3.37 + 10.3i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−11.73435826173288697080953354726, −11.12593140159748547042338669339, −9.981071370041678695984578539941, −8.764788302877526010052357295148, −7.60683441508504038057233170467, −6.29813143084175248283470030004, −5.70289555971433599391221499239, −5.22676758073681004735833534436, −4.20625873184832313936218728860, −2.87215165216740198008289743107,
1.22212848106144312048498453396, 2.29725242238346956754103001098, 4.00435841852346271444406577115, 4.94207718496964362912432172186, 5.91905070129823985517519546754, 6.51416881587981492667507526097, 7.68718683213960945908669266120, 9.649031047657457569622260901137, 10.51326108935585861412791420155, 11.07586241380524352841169300835