| L(s) = 1 | + (7.56 − 2.45i)2-s + (−12.6 − 9.16i)3-s + (−0.523 + 0.380i)4-s + (−67.3 + 207. i)5-s + (−118. − 38.3i)6-s + (−377. − 519. i)7-s + (−302. + 416. i)8-s + (75.0 + 231. i)9-s + 1.73e3i·10-s + (529. + 1.22e3i)11-s + 10.0·12-s + (−486. + 158. i)13-s + (−4.13e3 − 3.00e3i)14-s + (2.74e3 − 1.99e3i)15-s + (−1.25e3 + 3.85e3i)16-s + (−1.39e3 − 454. i)17-s + ⋯ |
| L(s) = 1 | + (0.946 − 0.307i)2-s + (−0.467 − 0.339i)3-s + (−0.00818 + 0.00594i)4-s + (−0.538 + 1.65i)5-s + (−0.546 − 0.177i)6-s + (−1.09 − 1.51i)7-s + (−0.590 + 0.813i)8-s + (0.103 + 0.317i)9-s + 1.73i·10-s + (0.397 + 0.917i)11-s + 0.00583·12-s + (−0.221 + 0.0719i)13-s + (−1.50 − 1.09i)14-s + (0.814 − 0.591i)15-s + (−0.305 + 0.941i)16-s + (−0.284 − 0.0925i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.678 - 0.735i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.678 - 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.280338 + 0.640027i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.280338 + 0.640027i\) |
| \(L(4)\) |
|
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 | \( 1 + (-529. - 1.22e3i)T \) |
| good | 2 | \( 1 + (-7.56 + 2.45i)T + (51.7 - 37.6i)T^{2} \) |
| 5 | \( 1 + (67.3 - 207. i)T + (-1.26e4 - 9.18e3i)T^{2} \) |
| 7 | \( 1 + (377. + 519. i)T + (-3.63e4 + 1.11e5i)T^{2} \) |
| 13 | \( 1 + (486. - 158. i)T + (3.90e6 - 2.83e6i)T^{2} \) |
| 17 | \( 1 + (1.39e3 + 454. i)T + (1.95e7 + 1.41e7i)T^{2} \) |
| 19 | \( 1 + (-3.21e3 + 4.42e3i)T + (-1.45e7 - 4.47e7i)T^{2} \) |
| 23 | \( 1 + 5.48e3T + 1.48e8T^{2} \) |
| 29 | \( 1 + (-2.04e3 - 2.81e3i)T + (-1.83e8 + 5.65e8i)T^{2} \) |
| 31 | \( 1 + (-7.01e3 - 2.15e4i)T + (-7.18e8 + 5.21e8i)T^{2} \) |
| 37 | \( 1 + (2.60e4 - 1.88e4i)T + (7.92e8 - 2.44e9i)T^{2} \) |
| 41 | \( 1 + (5.07e4 - 6.98e4i)T + (-1.46e9 - 4.51e9i)T^{2} \) |
| 43 | \( 1 + 2.35e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 + (-1.33e5 - 9.70e4i)T + (3.33e9 + 1.02e10i)T^{2} \) |
| 53 | \( 1 + (1.73e4 + 5.34e4i)T + (-1.79e10 + 1.30e10i)T^{2} \) |
| 59 | \( 1 + (1.32e5 - 9.62e4i)T + (1.30e10 - 4.01e10i)T^{2} \) |
| 61 | \( 1 + (-4.82e4 - 1.56e4i)T + (4.16e10 + 3.02e10i)T^{2} \) |
| 67 | \( 1 + 1.68e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + (8.13e4 - 2.50e5i)T + (-1.03e11 - 7.52e10i)T^{2} \) |
| 73 | \( 1 + (-2.86e5 - 3.94e5i)T + (-4.67e10 + 1.43e11i)T^{2} \) |
| 79 | \( 1 + (-7.31e5 + 2.37e5i)T + (1.96e11 - 1.42e11i)T^{2} \) |
| 83 | \( 1 + (8.61e5 + 2.79e5i)T + (2.64e11 + 1.92e11i)T^{2} \) |
| 89 | \( 1 + 6.32e5T + 4.96e11T^{2} \) |
| 97 | \( 1 + (2.41e5 + 7.42e5i)T + (-6.73e11 + 4.89e11i)T^{2} \) |
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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
−15.58694785089935119248298566962, −14.30802963749520496758430532679, −13.52511230053716639478671421219, −12.26755072914420640512823847330, −11.11607058145605277143664177104, −10.02146648848726576673881301826, −7.32836832094096397010908044572, −6.56731379554955138070011654684, −4.24955647882593029257505324286, −3.04156724918079426481317907494,
0.28072364063623919243558299518, 3.75453711483236959092680358037, 5.25707498264446337096625581168, 6.04394776911295506674536436618, 8.699799172527033350815784042876, 9.532300237107048587146359069899, 12.00993344213332295656466174038, 12.40858160333710301358600609563, 13.57639512289315774218611642207, 15.33064947592885836577135387380
