| L(s) = 1 | + (−1.32 − 0.503i)2-s + (−0.807 + 1.53i)3-s + (1.49 + 1.33i)4-s + (1.52 − 0.698i)5-s + (1.83 − 1.61i)6-s + (1.04 + 1.09i)7-s + (−1.30 − 2.51i)8-s + (−1.69 − 2.47i)9-s + (−2.37 + 0.152i)10-s + (−5.23 − 0.499i)11-s + (−3.24 + 1.21i)12-s + (0.612 + 0.118i)13-s + (−0.828 − 1.97i)14-s + (−0.164 + 2.90i)15-s + (0.456 + 3.97i)16-s + (−0.485 + 0.941i)17-s + ⋯ |
| L(s) = 1 | + (−0.934 − 0.356i)2-s + (−0.466 + 0.884i)3-s + (0.746 + 0.665i)4-s + (0.683 − 0.312i)5-s + (0.750 − 0.660i)6-s + (0.394 + 0.414i)7-s + (−0.460 − 0.887i)8-s + (−0.565 − 0.824i)9-s + (−0.750 + 0.0482i)10-s + (−1.57 − 0.150i)11-s + (−0.936 + 0.349i)12-s + (0.169 + 0.0327i)13-s + (−0.221 − 0.527i)14-s + (−0.0425 + 0.750i)15-s + (0.114 + 0.993i)16-s + (−0.117 + 0.228i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.204i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 - 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0273596 + 0.265032i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0273596 + 0.265032i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.32 + 0.503i)T \) |
| 3 | \( 1 + (0.807 - 1.53i)T \) |
| 67 | \( 1 + (7.46 + 3.34i)T \) |
| good | 5 | \( 1 + (-1.52 + 0.698i)T + (3.27 - 3.77i)T^{2} \) |
| 7 | \( 1 + (-1.04 - 1.09i)T + (-0.333 + 6.99i)T^{2} \) |
| 11 | \( 1 + (5.23 + 0.499i)T + (10.8 + 2.08i)T^{2} \) |
| 13 | \( 1 + (-0.612 - 0.118i)T + (12.0 + 4.83i)T^{2} \) |
| 17 | \( 1 + (0.485 - 0.941i)T + (-9.86 - 13.8i)T^{2} \) |
| 19 | \( 1 + (1.38 - 1.45i)T + (-0.904 - 18.9i)T^{2} \) |
| 23 | \( 1 + (6.96 - 2.78i)T + (16.6 - 15.8i)T^{2} \) |
| 29 | \( 1 + (0.764 + 0.441i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.0393 - 0.204i)T + (-28.7 + 11.5i)T^{2} \) |
| 37 | \( 1 + (1.01 + 1.76i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (4.04 - 0.192i)T + (40.8 - 3.89i)T^{2} \) |
| 43 | \( 1 + (-3.73 - 5.81i)T + (-17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + (0.0698 + 0.0549i)T + (11.0 + 45.6i)T^{2} \) |
| 53 | \( 1 + (2.65 - 4.12i)T + (-22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (-6.73 - 7.77i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (7.78 - 0.743i)T + (59.8 - 11.5i)T^{2} \) |
| 71 | \( 1 + (13.2 - 6.83i)T + (41.1 - 57.8i)T^{2} \) |
| 73 | \( 1 + (6.69 - 0.639i)T + (71.6 - 13.8i)T^{2} \) |
| 79 | \( 1 + (8.59 - 2.97i)T + (62.0 - 48.8i)T^{2} \) |
| 83 | \( 1 + (-4.64 + 6.52i)T + (-27.1 - 78.4i)T^{2} \) |
| 89 | \( 1 + (-1.14 + 0.165i)T + (85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-2.15 - 3.72i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.35046704814961372392930736083, −10.03405653789617444780353543354, −9.077988418992267773628058183860, −8.374905390425137584689933096801, −7.52055292169342908638319535365, −6.01725032417919601119673065596, −5.56978787511735938825324393871, −4.29764646780121001123945091627, −3.00317647953188136124108484846, −1.81434396174679870927829341021,
0.17425474403622200312091440835, 1.83049235848139639251039707709, 2.61437173307342458803617735343, 4.85617141137394957972692995845, 5.78965102930429275774313831438, 6.44847722327621810224946540752, 7.45096497164110028418288680656, 7.947307407694443301834883662956, 8.819274629972474279199019508259, 10.23485410509914815544476619557
