L(s) = 1 | + (−0.870 + 1.11i)2-s + (−0.483 − 1.94i)4-s + (1.71 − 0.988i)5-s + (2.27 + 1.35i)7-s + (2.58 + 1.15i)8-s + (−0.389 + 2.76i)10-s + (4.16 + 2.40i)11-s − 6.20i·13-s + (−3.48 + 1.35i)14-s + (−3.53 + 1.87i)16-s + (0.655 + 0.378i)17-s + (−3.65 − 6.33i)19-s + (−2.74 − 2.84i)20-s + (−6.30 + 2.54i)22-s + (−4.09 + 2.36i)23-s + ⋯ |
L(s) = 1 | + (−0.615 + 0.787i)2-s + (−0.241 − 0.970i)4-s + (0.765 − 0.441i)5-s + (0.858 + 0.512i)7-s + (0.913 + 0.407i)8-s + (−0.123 + 0.875i)10-s + (1.25 + 0.725i)11-s − 1.72i·13-s + (−0.932 + 0.361i)14-s + (−0.883 + 0.469i)16-s + (0.159 + 0.0918i)17-s + (−0.838 − 1.45i)19-s + (−0.613 − 0.635i)20-s + (−1.34 + 0.543i)22-s + (−0.854 + 0.493i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 - 0.348i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.937 - 0.348i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.42765 + 0.257145i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.42765 + 0.257145i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.870 - 1.11i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.27 - 1.35i)T \) |
good | 5 | \( 1 + (-1.71 + 0.988i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-4.16 - 2.40i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 6.20iT - 13T^{2} \) |
| 17 | \( 1 + (-0.655 - 0.378i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.65 + 6.33i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4.09 - 2.36i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 3.09T + 29T^{2} \) |
| 31 | \( 1 + (-3.85 + 6.67i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.98 - 6.90i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 6.55iT - 41T^{2} \) |
| 43 | \( 1 + 1.68iT - 43T^{2} \) |
| 47 | \( 1 + (1.30 + 2.25i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-7.08 + 12.2i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.705 + 1.22i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.09 - 2.93i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.74 - 2.73i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 13.5iT - 71T^{2} \) |
| 73 | \( 1 + (-4.77 - 2.75i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.07 + 1.77i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 1.69T + 83T^{2} \) |
| 89 | \( 1 + (3.63 - 2.09i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 2.34iT - 97T^{2} \) |
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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
−9.957926091049228950599286298472, −9.574177041486418365076171272708, −8.518382835939559169520607954032, −8.057315982216178371271445392364, −6.89122119729881614016552064586, −5.99225288151216205185712690952, −5.23651154425564574487241860761, −4.37299223604670444567648461590, −2.31570632335203946478592670293, −1.12199405189873752121959645519,
1.38448064780497727979481635921, 2.21189833934205229757521550550, 3.81482691609829248094561973507, 4.40964460575285760750079989069, 6.11630085616769913725892020380, 6.81685302188349480083179001863, 7.971081168882892545152290494941, 8.773789019378514551830770279951, 9.481906473741580371371559195885, 10.44053715940022546961827636090