| L(s) = 1 | + (5.30 + 9.99i)2-s + (81.6 − 33.8i)3-s + (−71.6 + 106. i)4-s + (132. − 319. i)5-s + (770. + 635. i)6-s + (381. + 381. i)7-s + (−1.44e3 − 152. i)8-s + (3.97e3 − 3.97e3i)9-s + (3.89e3 − 373. i)10-s + (−2.33e3 − 966. i)11-s + (−2.26e3 + 1.10e4i)12-s + (−404. − 976. i)13-s + (−1.78e3 + 5.83e3i)14-s − 3.05e4i·15-s + (−6.11e3 − 1.51e4i)16-s + 3.47e4i·17-s + ⋯ |
| L(s) = 1 | + (0.469 + 0.883i)2-s + (1.74 − 0.722i)3-s + (−0.559 + 0.828i)4-s + (0.473 − 1.14i)5-s + (1.45 + 1.20i)6-s + (0.420 + 0.420i)7-s + (−0.994 − 0.105i)8-s + (1.81 − 1.81i)9-s + (1.23 − 0.118i)10-s + (−0.528 − 0.218i)11-s + (−0.377 + 1.85i)12-s + (−0.0510 − 0.123i)13-s + (−0.173 + 0.568i)14-s − 2.33i·15-s + (−0.373 − 0.927i)16-s + 1.71i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 - 0.278i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.960 - 0.278i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(3.69315 + 0.524037i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.69315 + 0.524037i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-5.30 - 9.99i)T \) |
| good | 3 | \( 1 + (-81.6 + 33.8i)T + (1.54e3 - 1.54e3i)T^{2} \) |
| 5 | \( 1 + (-132. + 319. i)T + (-5.52e4 - 5.52e4i)T^{2} \) |
| 7 | \( 1 + (-381. - 381. i)T + 8.23e5iT^{2} \) |
| 11 | \( 1 + (2.33e3 + 966. i)T + (1.37e7 + 1.37e7i)T^{2} \) |
| 13 | \( 1 + (404. + 976. i)T + (-4.43e7 + 4.43e7i)T^{2} \) |
| 17 | \( 1 - 3.47e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 + (-1.09e4 - 2.65e4i)T + (-6.32e8 + 6.32e8i)T^{2} \) |
| 23 | \( 1 + (2.93e4 - 2.93e4i)T - 3.40e9iT^{2} \) |
| 29 | \( 1 + (5.24e4 - 2.17e4i)T + (1.21e10 - 1.21e10i)T^{2} \) |
| 31 | \( 1 + 3.12e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + (-1.60e5 + 3.87e5i)T + (-6.71e10 - 6.71e10i)T^{2} \) |
| 41 | \( 1 + (5.30e3 - 5.30e3i)T - 1.94e11iT^{2} \) |
| 43 | \( 1 + (-2.89e5 - 1.20e5i)T + (1.92e11 + 1.92e11i)T^{2} \) |
| 47 | \( 1 - 2.88e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 + (1.19e6 + 4.94e5i)T + (8.30e11 + 8.30e11i)T^{2} \) |
| 59 | \( 1 + (2.24e5 - 5.42e5i)T + (-1.75e12 - 1.75e12i)T^{2} \) |
| 61 | \( 1 + (-3.66e5 + 1.51e5i)T + (2.22e12 - 2.22e12i)T^{2} \) |
| 67 | \( 1 + (-2.71e6 + 1.12e6i)T + (4.28e12 - 4.28e12i)T^{2} \) |
| 71 | \( 1 + (1.18e6 + 1.18e6i)T + 9.09e12iT^{2} \) |
| 73 | \( 1 + (3.76e6 - 3.76e6i)T - 1.10e13iT^{2} \) |
| 79 | \( 1 + 9.34e5iT - 1.92e13T^{2} \) |
| 83 | \( 1 + (-2.62e6 - 6.34e6i)T + (-1.91e13 + 1.91e13i)T^{2} \) |
| 89 | \( 1 + (8.24e5 + 8.24e5i)T + 4.42e13iT^{2} \) |
| 97 | \( 1 - 1.07e7T + 8.07e13T^{2} \) |
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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
−14.96668353202175140424914086803, −14.20062511969454853214823103435, −12.96231907556233563671708666443, −12.63293792176148159474955000871, −9.371263156440535981114812899866, −8.453657684130279809688040357667, −7.68566959109375448925921182236, −5.71248675724690014008757490222, −3.76285419826153143032079280736, −1.80636501629798886812838820081,
2.25625398664111885322782524309, 3.16652258041028204791072869534, 4.71829592146215819803365826487, 7.39281534288536835581364833242, 9.180492377072420681367645141031, 10.08008131557702358769870775010, 11.09296273375646196682620078835, 13.27423164842788868377211065085, 14.09958226380522570510638585339, 14.66119548293633709301653121211
