L(s) = 1 | + (−1.93 + 0.489i)2-s + (−3.88 − 2.24i)3-s + (3.52 − 1.89i)4-s + (−0.133 + 0.231i)5-s + (8.63 + 2.44i)6-s + 7.24i·7-s + (−5.89 + 5.40i)8-s + (5.56 + 9.64i)9-s + (0.145 − 0.514i)10-s + 11.7i·11-s + (−17.9 − 0.519i)12-s + (4.17 + 7.22i)13-s + (−3.54 − 14.0i)14-s + (1.03 − 0.600i)15-s + (8.78 − 13.3i)16-s + (−7.11 + 12.3i)17-s + ⋯ |
L(s) = 1 | + (−0.969 + 0.244i)2-s + (−1.29 − 0.747i)3-s + (0.880 − 0.474i)4-s + (−0.0267 + 0.0463i)5-s + (1.43 + 0.408i)6-s + 1.03i·7-s + (−0.737 + 0.675i)8-s + (0.618 + 1.07i)9-s + (0.0145 − 0.0514i)10-s + 1.06i·11-s + (−1.49 − 0.0433i)12-s + (0.320 + 0.555i)13-s + (−0.253 − 1.00i)14-s + (0.0693 − 0.0400i)15-s + (0.549 − 0.835i)16-s + (−0.418 + 0.725i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0265 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0265 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.268820 + 0.276056i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.268820 + 0.276056i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.93 - 0.489i)T \) |
| 19 | \( 1 + (-11.9 + 14.7i)T \) |
good | 3 | \( 1 + (3.88 + 2.24i)T + (4.5 + 7.79i)T^{2} \) |
| 5 | \( 1 + (0.133 - 0.231i)T + (-12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 - 7.24iT - 49T^{2} \) |
| 11 | \( 1 - 11.7iT - 121T^{2} \) |
| 13 | \( 1 + (-4.17 - 7.22i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + (7.11 - 12.3i)T + (-144.5 - 250. i)T^{2} \) |
| 23 | \( 1 + (21.4 - 12.4i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (25.7 + 44.6i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 - 26.3iT - 961T^{2} \) |
| 37 | \( 1 + 20.1T + 1.36e3T^{2} \) |
| 41 | \( 1 + (38.5 - 66.8i)T + (-840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (30.0 + 17.3i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-31.9 + 18.4i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (8.75 + 15.1i)T + (-1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-82.8 - 47.8i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-26.2 - 45.3i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-11.8 + 6.86i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + (-39.7 - 22.9i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-20.0 + 34.7i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (106. + 61.2i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + 108. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (26.2 + 45.4i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-29.9 + 51.7i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−14.97489568206873919578035143634, −13.12571194172529551554423245953, −11.86405791842542884601835379181, −11.50659592437405554817870573439, −10.08952004363684278732651895820, −8.850481450406695520391763365467, −7.36529896733971485470073632394, −6.40333141509512835430284510174, −5.34081728992338719096848770396, −1.82909899905792982644583015734,
0.51992516672942714813786606603, 3.74336430207514866287306716890, 5.59256319662274036569281796490, 6.89779773245301103268330142000, 8.370561183991821661824789214190, 9.882746222346983321379308114575, 10.67580662105441235515980733401, 11.30554406164869125948752433230, 12.43098505579967710892673205753, 13.98635235188784667805321943426