| L(s) = 1 | − 11.0i·2-s + (−6.47 + 26.2i)3-s − 58.5·4-s + 94.9i·5-s + (290. + 71.6i)6-s + 96.5·7-s − 60.5i·8-s + (−645. − 339. i)9-s + 1.05e3·10-s − 430. i·11-s + (379. − 1.53e3i)12-s − 1.29e3·13-s − 1.06e3i·14-s + (−2.48e3 − 615. i)15-s − 4.41e3·16-s − 6.18e3i·17-s + ⋯ |
| L(s) = 1 | − 1.38i·2-s + (−0.239 + 0.970i)3-s − 0.914·4-s + 0.759i·5-s + (1.34 + 0.331i)6-s + 0.281·7-s − 0.118i·8-s + (−0.884 − 0.465i)9-s + 1.05·10-s − 0.323i·11-s + (0.219 − 0.887i)12-s − 0.587·13-s − 0.389i·14-s + (−0.737 − 0.182i)15-s − 1.07·16-s − 1.25i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 - 0.239i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.970 - 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.0540134 + 0.443824i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0540134 + 0.443824i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (6.47 - 26.2i)T \) |
| 23 | \( 1 + 2.53e3iT \) |
| good | 2 | \( 1 + 11.0iT - 64T^{2} \) |
| 5 | \( 1 - 94.9iT - 1.56e4T^{2} \) |
| 7 | \( 1 - 96.5T + 1.17e5T^{2} \) |
| 11 | \( 1 + 430. iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 1.29e3T + 4.82e6T^{2} \) |
| 17 | \( 1 + 6.18e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 1.04e4T + 4.70e7T^{2} \) |
| 29 | \( 1 + 707. iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 2.11e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + 8.09e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + 2.22e3iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 2.12e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + 1.32e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 3.89e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 2.40e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 4.91e4T + 5.15e10T^{2} \) |
| 67 | \( 1 - 3.18e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 3.32e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 1.76e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 1.06e5T + 2.43e11T^{2} \) |
| 83 | \( 1 - 3.87e4iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 7.21e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 7.49e5T + 8.32e11T^{2} \) |
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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
−12.51871763724703145787795770193, −11.44193717970809310157611774709, −10.77176558060219835514275372745, −9.955939457368591771755477820348, −8.811545733530889925698607592863, −6.77386942870072152467585077683, −4.92679522406522913038243047252, −3.57560538783505491727537289355, −2.40522391734225836201013025997, −0.16947650317124677311446018264,
1.84484949979575099908288315753, 4.73508708439567831051562385669, 5.90058362160714714786281425037, 6.96780682661990665760210704664, 8.070472384685062226603163642662, 8.836829252819702795199457253619, 10.84694447708438638913133266966, 12.31594559165513122173242224113, 13.05641651171699202592768720436, 14.33439354632587780596093543924
