| L(s) = 1 | + (−0.306 + 1.38i)2-s + (−1.12 − 1.31i)3-s + (−1.81 − 0.846i)4-s + (−0.193 + 0.974i)5-s + (2.16 − 1.15i)6-s + (3.70 + 1.53i)7-s + (1.72 − 2.24i)8-s + (−0.460 + 2.96i)9-s + (−1.28 − 0.566i)10-s + (2.91 + 4.35i)11-s + (0.929 + 3.33i)12-s + (−4.94 + 0.984i)13-s + (−3.25 + 4.64i)14-s + (1.50 − 0.843i)15-s + (2.56 + 3.06i)16-s + (0.683 + 0.683i)17-s + ⋯ |
| L(s) = 1 | + (−0.216 + 0.976i)2-s + (−0.650 − 0.759i)3-s + (−0.906 − 0.423i)4-s + (−0.0867 + 0.435i)5-s + (0.882 − 0.470i)6-s + (1.40 + 0.580i)7-s + (0.609 − 0.792i)8-s + (−0.153 + 0.988i)9-s + (−0.406 − 0.179i)10-s + (0.877 + 1.31i)11-s + (0.268 + 0.963i)12-s + (−1.37 + 0.273i)13-s + (−0.870 + 1.24i)14-s + (0.387 − 0.217i)15-s + (0.641 + 0.766i)16-s + (0.165 + 0.165i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.111 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.111 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.649245 + 0.580711i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.649245 + 0.580711i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.306 - 1.38i)T \) |
| 3 | \( 1 + (1.12 + 1.31i)T \) |
| good | 5 | \( 1 + (0.193 - 0.974i)T + (-4.61 - 1.91i)T^{2} \) |
| 7 | \( 1 + (-3.70 - 1.53i)T + (4.94 + 4.94i)T^{2} \) |
| 11 | \( 1 + (-2.91 - 4.35i)T + (-4.20 + 10.1i)T^{2} \) |
| 13 | \( 1 + (4.94 - 0.984i)T + (12.0 - 4.97i)T^{2} \) |
| 17 | \( 1 + (-0.683 - 0.683i)T + 17iT^{2} \) |
| 19 | \( 1 + (-2.19 + 0.437i)T + (17.5 - 7.27i)T^{2} \) |
| 23 | \( 1 + (-5.93 + 2.45i)T + (16.2 - 16.2i)T^{2} \) |
| 29 | \( 1 + (3.04 + 2.03i)T + (11.0 + 26.7i)T^{2} \) |
| 31 | \( 1 + 6.01T + 31T^{2} \) |
| 37 | \( 1 + (1.17 - 5.89i)T + (-34.1 - 14.1i)T^{2} \) |
| 41 | \( 1 + (2.28 + 5.50i)T + (-28.9 + 28.9i)T^{2} \) |
| 43 | \( 1 + (4.39 + 6.57i)T + (-16.4 + 39.7i)T^{2} \) |
| 47 | \( 1 + (0.410 - 0.410i)T - 47iT^{2} \) |
| 53 | \( 1 + (-3.47 + 2.31i)T + (20.2 - 48.9i)T^{2} \) |
| 59 | \( 1 + (0.669 + 0.133i)T + (54.5 + 22.5i)T^{2} \) |
| 61 | \( 1 + (1.32 + 0.885i)T + (23.3 + 56.3i)T^{2} \) |
| 67 | \( 1 + (-5.83 + 8.72i)T + (-25.6 - 61.8i)T^{2} \) |
| 71 | \( 1 + (2.48 - 6.00i)T + (-50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (0.0562 + 0.135i)T + (-51.6 + 51.6i)T^{2} \) |
| 79 | \( 1 + (-0.732 + 0.732i)T - 79iT^{2} \) |
| 83 | \( 1 + (-1.38 - 6.96i)T + (-76.6 + 31.7i)T^{2} \) |
| 89 | \( 1 + (3.52 - 8.51i)T + (-62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + 7.20iT - 97T^{2} \) |
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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
−12.64589306880414476922876541080, −11.92563186746129931248064441673, −10.89063539264891194267850326163, −9.628162831081147395354992477256, −8.474725819203451379566193860338, −7.24633118831223929035741138056, −6.97566279254779756223850352571, −5.35167967208306494355333747187, −4.70552140780635635914374848187, −1.81265657959087075277140670661,
1.05240816910252972344120895873, 3.41461876649948311089359587031, 4.66847282303523253570818415545, 5.35408720631855344922031249139, 7.41584732539198098070151653942, 8.665763952771497001715161884854, 9.453720479943826125407107898349, 10.62691497353025454883528561203, 11.30663698747196077498824611694, 11.87136290596814550086508766519
