| L(s) = 1 | + (11.0 + 11.0i)3-s + (−47.0 − 47.0i)5-s − 400.·7-s + 242. i·9-s + (1.04e3 − 1.04e3i)11-s + (−642. + 642. i)13-s − 1.03e3i·15-s + 1.91e3·17-s + (55.8 + 55.8i)19-s + (−4.41e3 − 4.41e3i)21-s + 4.19e3·23-s − 1.11e4i·25-s + (−2.67e3 + 2.67e3i)27-s + (1.56e4 − 1.56e4i)29-s − 3.62e3i·31-s + ⋯ |
| L(s) = 1 | + (0.408 + 0.408i)3-s + (−0.376 − 0.376i)5-s − 1.16·7-s + 0.333i·9-s + (0.784 − 0.784i)11-s + (−0.292 + 0.292i)13-s − 0.307i·15-s + 0.389·17-s + (0.00814 + 0.00814i)19-s + (−0.476 − 0.476i)21-s + 0.344·23-s − 0.716i·25-s + (−0.136 + 0.136i)27-s + (0.641 − 0.641i)29-s − 0.121i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.439 - 0.898i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.439 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.9684530159\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9684530159\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-11.0 - 11.0i)T \) |
| good | 5 | \( 1 + (47.0 + 47.0i)T + 1.56e4iT^{2} \) |
| 7 | \( 1 + 400.T + 1.17e5T^{2} \) |
| 11 | \( 1 + (-1.04e3 + 1.04e3i)T - 1.77e6iT^{2} \) |
| 13 | \( 1 + (642. - 642. i)T - 4.82e6iT^{2} \) |
| 17 | \( 1 - 1.91e3T + 2.41e7T^{2} \) |
| 19 | \( 1 + (-55.8 - 55.8i)T + 4.70e7iT^{2} \) |
| 23 | \( 1 - 4.19e3T + 1.48e8T^{2} \) |
| 29 | \( 1 + (-1.56e4 + 1.56e4i)T - 5.94e8iT^{2} \) |
| 31 | \( 1 + 3.62e3iT - 8.87e8T^{2} \) |
| 37 | \( 1 + (3.14e4 + 3.14e4i)T + 2.56e9iT^{2} \) |
| 41 | \( 1 - 8.38e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + (9.50e4 - 9.50e4i)T - 6.32e9iT^{2} \) |
| 47 | \( 1 + 8.18e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 + (-1.54e5 - 1.54e5i)T + 2.21e10iT^{2} \) |
| 59 | \( 1 + (2.41e5 - 2.41e5i)T - 4.21e10iT^{2} \) |
| 61 | \( 1 + (2.85e5 - 2.85e5i)T - 5.15e10iT^{2} \) |
| 67 | \( 1 + (-3.07e5 - 3.07e5i)T + 9.04e10iT^{2} \) |
| 71 | \( 1 - 4.95e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + 6.06e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 - 6.50e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + (1.57e5 + 1.57e5i)T + 3.26e11iT^{2} \) |
| 89 | \( 1 + 4.04e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 6.80e5T + 8.32e11T^{2} \) |
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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
−10.44309402732194968205571868060, −9.616166193289853096642191008784, −8.896938812876773178707201593576, −8.028233405260003848882790288507, −6.80261217404168177656868503015, −5.93571040582134892066987298622, −4.56907423657769140737885817235, −3.63596055453172167618956823759, −2.73241491982593668383264708700, −1.01710262317823077090010591606,
0.23280731125887236763701849674, 1.67696734553086861646517037164, 3.04329450359758392497946055730, 3.72911646065314938620191805666, 5.19943217575681210339162473253, 6.65996445182807927395142885126, 6.98323994396235597490853072554, 8.122521469656747040443842000151, 9.233507375118591513633517284770, 9.871173381529299979537179763893
