| L(s) = 1 | + (−23.2 − 23.2i)3-s + (−41.0 − 41.0i)5-s + 327.·7-s + 354. i·9-s + (266. − 266. i)11-s + (−1.20e3 + 1.20e3i)13-s + 1.91e3i·15-s + 36.0·17-s + (−7.73e3 − 7.73e3i)19-s + (−7.61e3 − 7.61e3i)21-s + 8.60e3·23-s − 1.22e4i·25-s + (−8.72e3 + 8.72e3i)27-s + (1.92e4 − 1.92e4i)29-s + 56.8i·31-s + ⋯ |
| L(s) = 1 | + (−0.861 − 0.861i)3-s + (−0.328 − 0.328i)5-s + 0.953·7-s + 0.485i·9-s + (0.199 − 0.199i)11-s + (−0.546 + 0.546i)13-s + 0.565i·15-s + 0.00732·17-s + (−1.12 − 1.12i)19-s + (−0.821 − 0.821i)21-s + 0.707·23-s − 0.784i·25-s + (−0.443 + 0.443i)27-s + (0.791 − 0.791i)29-s + 0.00190i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.1581784002\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1581784002\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + (23.2 + 23.2i)T + 729iT^{2} \) |
| 5 | \( 1 + (41.0 + 41.0i)T + 1.56e4iT^{2} \) |
| 7 | \( 1 - 327.T + 1.17e5T^{2} \) |
| 11 | \( 1 + (-266. + 266. i)T - 1.77e6iT^{2} \) |
| 13 | \( 1 + (1.20e3 - 1.20e3i)T - 4.82e6iT^{2} \) |
| 17 | \( 1 - 36.0T + 2.41e7T^{2} \) |
| 19 | \( 1 + (7.73e3 + 7.73e3i)T + 4.70e7iT^{2} \) |
| 23 | \( 1 - 8.60e3T + 1.48e8T^{2} \) |
| 29 | \( 1 + (-1.92e4 + 1.92e4i)T - 5.94e8iT^{2} \) |
| 31 | \( 1 - 56.8iT - 8.87e8T^{2} \) |
| 37 | \( 1 + (6.12e4 + 6.12e4i)T + 2.56e9iT^{2} \) |
| 41 | \( 1 + 8.59e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + (5.85e4 - 5.85e4i)T - 6.32e9iT^{2} \) |
| 47 | \( 1 - 1.57e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 + (-1.99e4 - 1.99e4i)T + 2.21e10iT^{2} \) |
| 59 | \( 1 + (1.86e4 - 1.86e4i)T - 4.21e10iT^{2} \) |
| 61 | \( 1 + (-2.65e3 + 2.65e3i)T - 5.15e10iT^{2} \) |
| 67 | \( 1 + (-1.18e5 - 1.18e5i)T + 9.04e10iT^{2} \) |
| 71 | \( 1 + 5.22e5T + 1.28e11T^{2} \) |
| 73 | \( 1 - 4.32e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 - 3.74e4iT - 2.43e11T^{2} \) |
| 83 | \( 1 + (-7.55e5 - 7.55e5i)T + 3.26e11iT^{2} \) |
| 89 | \( 1 - 1.19e6iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 1.01e6T + 8.32e11T^{2} \) |
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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
−10.69050111201651002296277917792, −9.164319248944449197434045020754, −8.244437263212456043085815682495, −7.17216125145755629588351420560, −6.39468180816715965529740834192, −5.15626835456775845333176437867, −4.26504340436693519922073372629, −2.29906900994007731470437005523, −1.07344265259485954669665202831, −0.05058436137257660360681002238,
1.67503937448638761765260520064, 3.40844114512276503886902469255, 4.66246505268298915297605487973, 5.25630110647629216087934558260, 6.54602369956894816479350195638, 7.74531766980455453861695204834, 8.712222564839488461644446845567, 10.15994036486364544234139225440, 10.56408346509489908696052042252, 11.52068948681562627796288978888
