| 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
−11.87136290596814550086508766519, −11.30663698747196077498824611694, −10.62691497353025454883528561203, −9.453720479943826125407107898349, −8.665763952771497001715161884854, −7.41584732539198098070151653942, −5.35408720631855344922031249139, −4.66847282303523253570818415545, −3.41461876649948311089359587031, −1.05240816910252972344120895873,
1.81265657959087075277140670661, 4.70552140780635635914374848187, 5.35167967208306494355333747187, 6.97566279254779756223850352571, 7.24633118831223929035741138056, 8.474725819203451379566193860338, 9.628162831081147395354992477256, 10.89063539264891194267850326163, 11.92563186746129931248064441673, 12.64589306880414476922876541080
