| L(s) = 1 | + (−1.32 − 1.11i)3-s + (0.102 − 0.712i)5-s + (0.0729 + 0.763i)7-s + (0.497 + 2.95i)9-s + (−0.457 + 0.183i)11-s + (3.42 + 3.59i)13-s + (−0.932 + 0.827i)15-s + (−6.00 + 2.07i)17-s + (1.10 + 0.105i)19-s + (0.757 − 1.09i)21-s + (7.01 − 0.334i)23-s + (4.30 + 1.26i)25-s + (2.65 − 4.46i)27-s + (−7.80 + 4.50i)29-s + (2.71 − 2.84i)31-s + ⋯ |
| L(s) = 1 | + (−0.763 − 0.645i)3-s + (0.0457 − 0.318i)5-s + (0.0275 + 0.288i)7-s + (0.165 + 0.986i)9-s + (−0.138 + 0.0552i)11-s + (0.950 + 0.996i)13-s + (−0.240 + 0.213i)15-s + (−1.45 + 0.504i)17-s + (0.254 + 0.0243i)19-s + (0.165 − 0.238i)21-s + (1.46 − 0.0697i)23-s + (0.860 + 0.252i)25-s + (0.510 − 0.860i)27-s + (−1.45 + 0.837i)29-s + (0.487 − 0.511i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.232i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 - 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.13980 + 0.134266i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.13980 + 0.134266i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.32 + 1.11i)T \) |
| 67 | \( 1 + (0.562 + 8.16i)T \) |
| good | 5 | \( 1 + (-0.102 + 0.712i)T + (-4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (-0.0729 - 0.763i)T + (-6.87 + 1.32i)T^{2} \) |
| 11 | \( 1 + (0.457 - 0.183i)T + (7.96 - 7.59i)T^{2} \) |
| 13 | \( 1 + (-3.42 - 3.59i)T + (-0.618 + 12.9i)T^{2} \) |
| 17 | \( 1 + (6.00 - 2.07i)T + (13.3 - 10.5i)T^{2} \) |
| 19 | \( 1 + (-1.10 - 0.105i)T + (18.6 + 3.59i)T^{2} \) |
| 23 | \( 1 + (-7.01 + 0.334i)T + (22.8 - 2.18i)T^{2} \) |
| 29 | \( 1 + (7.80 - 4.50i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.71 + 2.84i)T + (-1.47 - 30.9i)T^{2} \) |
| 37 | \( 1 + (-0.597 + 1.03i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-8.84 - 1.70i)T + (38.0 + 15.2i)T^{2} \) |
| 43 | \( 1 + (-3.66 - 3.17i)T + (6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + (-2.65 + 5.15i)T + (-27.2 - 38.2i)T^{2} \) |
| 53 | \( 1 + (-1.01 - 1.17i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-1.87 - 6.39i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (2.71 - 6.77i)T + (-44.1 - 42.0i)T^{2} \) |
| 71 | \( 1 + (8.12 + 2.81i)T + (55.8 + 43.8i)T^{2} \) |
| 73 | \( 1 + (-10.8 - 4.34i)T + (52.8 + 50.3i)T^{2} \) |
| 79 | \( 1 + (-7.69 + 1.86i)T + (70.2 - 36.1i)T^{2} \) |
| 83 | \( 1 + (9.04 - 11.5i)T + (-19.5 - 80.6i)T^{2} \) |
| 89 | \( 1 + (-2.00 + 3.12i)T + (-36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (-1.81 - 1.04i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.68860938539424386466249343828, −9.130732467560252126168232313456, −8.804062384223774634065064231151, −7.53295682012490402935221673204, −6.75886169669362076542045589006, −5.98275853742768569757681274652, −5.03802337238035862227040602412, −4.08417438756907468199722333917, −2.39689920442145536168236576157, −1.19038406408309969671176412524,
0.76216128753804080817235999833, 2.80043722965009042220111724913, 3.90061915897773914636834855014, 4.87690488710913331634839284245, 5.78371525332162912970368361629, 6.63788190853048832876334221252, 7.50303983281002787186911993010, 8.780940949780885279814155581273, 9.393738776856631185759685088540, 10.55187459161727361860583645426
