| L(s) = 1 | + (−0.674 + 0.219i)2-s + (12.6 + 9.16i)3-s + (−51.3 + 37.3i)4-s + (−8.69 + 26.7i)5-s + (−10.5 − 3.41i)6-s + (−301. − 415. i)7-s + (53.1 − 73.1i)8-s + (75.0 + 231. i)9-s − 19.9i·10-s + (−1.32e3 − 75.6i)11-s − 989.·12-s + (−212. + 69.0i)13-s + (294. + 214. i)14-s + (−354. + 257. i)15-s + (1.23e3 − 3.80e3i)16-s + (−7.42e3 − 2.41e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.0843 + 0.0273i)2-s + (0.467 + 0.339i)3-s + (−0.802 + 0.583i)4-s + (−0.0695 + 0.214i)5-s + (−0.0486 − 0.0158i)6-s + (−0.880 − 1.21i)7-s + (0.103 − 0.142i)8-s + (0.103 + 0.317i)9-s − 0.0199i·10-s + (−0.998 − 0.0568i)11-s − 0.572·12-s + (−0.0967 + 0.0314i)13-s + (0.107 + 0.0780i)14-s + (−0.105 + 0.0764i)15-s + (0.301 − 0.928i)16-s + (−1.51 − 0.491i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.961 + 0.275i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.961 + 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.0157452 - 0.112030i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0157452 - 0.112030i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-12.6 - 9.16i)T \) |
| 11 | \( 1 + (1.32e3 + 75.6i)T \) |
| good | 2 | \( 1 + (0.674 - 0.219i)T + (51.7 - 37.6i)T^{2} \) |
| 5 | \( 1 + (8.69 - 26.7i)T + (-1.26e4 - 9.18e3i)T^{2} \) |
| 7 | \( 1 + (301. + 415. i)T + (-3.63e4 + 1.11e5i)T^{2} \) |
| 13 | \( 1 + (212. - 69.0i)T + (3.90e6 - 2.83e6i)T^{2} \) |
| 17 | \( 1 + (7.42e3 + 2.41e3i)T + (1.95e7 + 1.41e7i)T^{2} \) |
| 19 | \( 1 + (7.54e3 - 1.03e4i)T + (-1.45e7 - 4.47e7i)T^{2} \) |
| 23 | \( 1 - 1.71e4T + 1.48e8T^{2} \) |
| 29 | \( 1 + (1.40e4 + 1.93e4i)T + (-1.83e8 + 5.65e8i)T^{2} \) |
| 31 | \( 1 + (-9.59e3 - 2.95e4i)T + (-7.18e8 + 5.21e8i)T^{2} \) |
| 37 | \( 1 + (-3.18e4 + 2.31e4i)T + (7.92e8 - 2.44e9i)T^{2} \) |
| 41 | \( 1 + (5.00e4 - 6.89e4i)T + (-1.46e9 - 4.51e9i)T^{2} \) |
| 43 | \( 1 - 3.27e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 + (1.30e5 + 9.47e4i)T + (3.33e9 + 1.02e10i)T^{2} \) |
| 53 | \( 1 + (2.57e4 + 7.91e4i)T + (-1.79e10 + 1.30e10i)T^{2} \) |
| 59 | \( 1 + (1.32e5 - 9.65e4i)T + (1.30e10 - 4.01e10i)T^{2} \) |
| 61 | \( 1 + (245. + 79.6i)T + (4.16e10 + 3.02e10i)T^{2} \) |
| 67 | \( 1 - 1.18e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + (1.25e4 - 3.84e4i)T + (-1.03e11 - 7.52e10i)T^{2} \) |
| 73 | \( 1 + (9.58e4 + 1.31e5i)T + (-4.67e10 + 1.43e11i)T^{2} \) |
| 79 | \( 1 + (3.16e5 - 1.02e5i)T + (1.96e11 - 1.42e11i)T^{2} \) |
| 83 | \( 1 + (2.17e5 + 7.05e4i)T + (2.64e11 + 1.92e11i)T^{2} \) |
| 89 | \( 1 - 8.47e5T + 4.96e11T^{2} \) |
| 97 | \( 1 + (3.26e5 + 1.00e6i)T + (-6.73e11 + 4.89e11i)T^{2} \) |
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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
−16.25545310281797409225237369747, −14.81312331837860074487879709304, −13.40942568267704834541071328855, −12.96036231166140337901327152132, −10.78968580195762855152895190795, −9.701620396302474700993824216035, −8.350720461967746640698906258079, −7.01117182017090453358779664652, −4.54648116907043528686036280635, −3.21600841504191870138839708908,
0.05568361578772772856010470292, 2.52840594036160766811596965061, 4.86435448216521885231999696419, 6.48323802596721733728556002889, 8.613900232869337336244751140237, 9.236140536538075545975750940854, 10.83992281670549774249602694641, 12.90349982398992607873064007294, 13.15402415572711674183359991174, 14.98348823756547066773544955241
