| L(s) = 1 | + (−2.25 − 3.90i)2-s + (21.8 − 37.8i)4-s + (−53.9 + 31.1i)5-s + (218. − 264. i)7-s − 485.·8-s + (243. + 140. i)10-s + (9.71 − 16.8i)11-s − 1.64e3i·13-s + (−1.52e3 − 258. i)14-s + (−304. − 527. i)16-s + (186. + 107. i)17-s + (−8.23e3 + 4.75e3i)19-s + 2.72e3i·20-s − 87.5·22-s + (−7.22e3 − 1.25e4i)23-s + ⋯ |
| L(s) = 1 | + (−0.281 − 0.487i)2-s + (0.341 − 0.591i)4-s + (−0.431 + 0.249i)5-s + (0.637 − 0.770i)7-s − 0.947·8-s + (0.243 + 0.140i)10-s + (0.00730 − 0.0126i)11-s − 0.747i·13-s + (−0.555 − 0.0943i)14-s + (−0.0743 − 0.128i)16-s + (0.0378 + 0.0218i)17-s + (−1.20 + 0.693i)19-s + 0.340i·20-s − 0.00822·22-s + (−0.593 − 1.02i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0415i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.999 - 0.0415i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.0179434 + 0.863711i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0179434 + 0.863711i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + (-218. + 264. i)T \) |
| good | 2 | \( 1 + (2.25 + 3.90i)T + (-32 + 55.4i)T^{2} \) |
| 5 | \( 1 + (53.9 - 31.1i)T + (7.81e3 - 1.35e4i)T^{2} \) |
| 11 | \( 1 + (-9.71 + 16.8i)T + (-8.85e5 - 1.53e6i)T^{2} \) |
| 13 | \( 1 + 1.64e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + (-186. - 107. i)T + (1.20e7 + 2.09e7i)T^{2} \) |
| 19 | \( 1 + (8.23e3 - 4.75e3i)T + (2.35e7 - 4.07e7i)T^{2} \) |
| 23 | \( 1 + (7.22e3 + 1.25e4i)T + (-7.40e7 + 1.28e8i)T^{2} \) |
| 29 | \( 1 + 4.30e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + (7.79e3 + 4.50e3i)T + (4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + (-1.65e4 - 2.86e4i)T + (-1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 + 7.37e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 4.76e3T + 6.32e9T^{2} \) |
| 47 | \( 1 + (-6.37e4 + 3.68e4i)T + (5.38e9 - 9.33e9i)T^{2} \) |
| 53 | \( 1 + (-1.19e5 + 2.06e5i)T + (-1.10e10 - 1.91e10i)T^{2} \) |
| 59 | \( 1 + (-2.82e5 - 1.63e5i)T + (2.10e10 + 3.65e10i)T^{2} \) |
| 61 | \( 1 + (-3.50e5 + 2.02e5i)T + (2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (1.09e5 - 1.90e5i)T + (-4.52e10 - 7.83e10i)T^{2} \) |
| 71 | \( 1 + 3.50e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + (1.74e5 + 1.00e5i)T + (7.56e10 + 1.31e11i)T^{2} \) |
| 79 | \( 1 + (1.97e5 + 3.42e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 - 1.32e4iT - 3.26e11T^{2} \) |
| 89 | \( 1 + (-1.99e5 + 1.15e5i)T + (2.48e11 - 4.30e11i)T^{2} \) |
| 97 | \( 1 - 6.62e5iT - 8.32e11T^{2} \) |
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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
−13.05174628290841652866065433078, −11.71330204327838130416512247480, −10.80212067911446548168852033923, −10.06945700637138601491496907954, −8.446507353352076545965113963728, −7.16146480400035498572065987488, −5.66078693311674411328784975524, −3.86007482302288796605811666622, −1.98449752174967151408048713275, −0.36114864621370906664187662365,
2.21515898145143059941719260838, 4.10505951126567602353563704921, 5.86131044555329052050656446237, 7.29219666001990754768017955867, 8.346794905038447001213664274602, 9.245958749425552889473514738157, 11.22723118407699105028353076263, 11.92755881429585466563244206504, 13.03192791821788974976912096942, 14.63262235412492977247747868236
