| L(s) = 1 | + (−6.49 + 11.2i)2-s + (−40.4 + 23.3i)3-s + (−52.3 − 90.7i)4-s + (26.6 + 15.3i)5-s − 606. i·6-s + 530.·8-s + (723. − 1.25e3i)9-s + (−346. + 200. i)10-s + (−726. − 1.25e3i)11-s + (4.23e3 + 2.44e3i)12-s + 1.87e3i·13-s − 1.43e3·15-s + (−89.8 + 155. i)16-s + (−5.32e3 + 3.07e3i)17-s + (9.40e3 + 1.62e4i)18-s + (−1.53e3 − 883. i)19-s + ⋯ |
| L(s) = 1 | + (−0.812 + 1.40i)2-s + (−1.49 + 0.863i)3-s + (−0.818 − 1.41i)4-s + (0.213 + 0.123i)5-s − 2.80i·6-s + 1.03·8-s + (0.992 − 1.71i)9-s + (−0.346 + 0.200i)10-s + (−0.545 − 0.945i)11-s + (2.45 + 1.41i)12-s + 0.853i·13-s − 0.425·15-s + (−0.0219 + 0.0380i)16-s + (−1.08 + 0.625i)17-s + (1.61 + 2.79i)18-s + (−0.223 − 0.128i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.580 - 0.814i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.580 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.300913 + 0.155013i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.300913 + 0.155013i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 + (6.49 - 11.2i)T + (-32 - 55.4i)T^{2} \) |
| 3 | \( 1 + (40.4 - 23.3i)T + (364.5 - 631. i)T^{2} \) |
| 5 | \( 1 + (-26.6 - 15.3i)T + (7.81e3 + 1.35e4i)T^{2} \) |
| 11 | \( 1 + (726. + 1.25e3i)T + (-8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 - 1.87e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + (5.32e3 - 3.07e3i)T + (1.20e7 - 2.09e7i)T^{2} \) |
| 19 | \( 1 + (1.53e3 + 883. i)T + (2.35e7 + 4.07e7i)T^{2} \) |
| 23 | \( 1 + (4.85e3 - 8.41e3i)T + (-7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 - 4.00e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + (-2.78e4 + 1.60e4i)T + (4.43e8 - 7.68e8i)T^{2} \) |
| 37 | \( 1 + (-1.88e4 + 3.27e4i)T + (-1.28e9 - 2.22e9i)T^{2} \) |
| 41 | \( 1 - 3.70e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 4.43e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + (7.36e4 + 4.25e4i)T + (5.38e9 + 9.33e9i)T^{2} \) |
| 53 | \( 1 + (-5.79e4 - 1.00e5i)T + (-1.10e10 + 1.91e10i)T^{2} \) |
| 59 | \( 1 + (1.99e4 - 1.15e4i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (-9.05e4 - 5.22e4i)T + (2.57e10 + 4.46e10i)T^{2} \) |
| 67 | \( 1 + (944. + 1.63e3i)T + (-4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 + 6.70e4T + 1.28e11T^{2} \) |
| 73 | \( 1 + (-6.43e5 + 3.71e5i)T + (7.56e10 - 1.31e11i)T^{2} \) |
| 79 | \( 1 + (-1.38e5 + 2.39e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + 6.75e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + (-6.08e5 - 3.51e5i)T + (2.48e11 + 4.30e11i)T^{2} \) |
| 97 | \( 1 + 5.43e5iT - 8.32e11T^{2} \) |
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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
−15.22158285221870318845250024193, −13.74491438654882346547502142656, −11.82428663441528631776194784010, −10.67411115391016774628599879095, −9.688051526840416483914669937072, −8.375969314784213742241171283134, −6.56572252603174949364239301401, −5.93385609011489574592506279442, −4.57461694343449736631550097087, −0.33976194082404815179629029812,
0.876863900771799383182476237285, 2.30433099026749207024846234142, 4.91742679451285667375185916323, 6.61231307446315265614752312440, 8.146516386595804839955512532776, 9.927500590216716053006795353085, 10.73739815369411284196527114535, 11.77104704773355187442153619933, 12.54354686509347305326962878969, 13.33214937296889203635315194328
