| L(s) = 1 | + (−7.35 − 8.59i)2-s + (−46.0 + 19.0i)3-s + (−19.8 + 126. i)4-s + (−76.0 + 183. i)5-s + (502. + 255. i)6-s + (60.8 + 60.8i)7-s + (1.23e3 − 758. i)8-s + (211. − 211. i)9-s + (2.13e3 − 696. i)10-s + (−2.10e3 − 873. i)11-s + (−1.49e3 − 6.20e3i)12-s + (−817. − 1.97e3i)13-s + (75.8 − 971. i)14-s − 9.91e3i·15-s + (−1.55e4 − 5.02e3i)16-s − 4.84e3i·17-s + ⋯ |
| L(s) = 1 | + (−0.649 − 0.760i)2-s + (−0.984 + 0.407i)3-s + (−0.155 + 0.987i)4-s + (−0.272 + 0.657i)5-s + (0.950 + 0.483i)6-s + (0.0670 + 0.0670i)7-s + (0.851 − 0.523i)8-s + (0.0966 − 0.0966i)9-s + (0.676 − 0.220i)10-s + (−0.477 − 0.197i)11-s + (−0.250 − 1.03i)12-s + (−0.103 − 0.249i)13-s + (0.00739 − 0.0945i)14-s − 0.758i·15-s + (−0.951 − 0.306i)16-s − 0.238i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.192 + 0.981i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.192 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.272816 - 0.331492i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.272816 - 0.331492i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (7.35 + 8.59i)T \) |
| good | 3 | \( 1 + (46.0 - 19.0i)T + (1.54e3 - 1.54e3i)T^{2} \) |
| 5 | \( 1 + (76.0 - 183. i)T + (-5.52e4 - 5.52e4i)T^{2} \) |
| 7 | \( 1 + (-60.8 - 60.8i)T + 8.23e5iT^{2} \) |
| 11 | \( 1 + (2.10e3 + 873. i)T + (1.37e7 + 1.37e7i)T^{2} \) |
| 13 | \( 1 + (817. + 1.97e3i)T + (-4.43e7 + 4.43e7i)T^{2} \) |
| 17 | \( 1 + 4.84e3iT - 4.10e8T^{2} \) |
| 19 | \( 1 + (1.04e4 + 2.53e4i)T + (-6.32e8 + 6.32e8i)T^{2} \) |
| 23 | \( 1 + (-6.67e4 + 6.67e4i)T - 3.40e9iT^{2} \) |
| 29 | \( 1 + (-1.20e4 + 4.97e3i)T + (1.21e10 - 1.21e10i)T^{2} \) |
| 31 | \( 1 - 1.38e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + (1.19e5 - 2.87e5i)T + (-6.71e10 - 6.71e10i)T^{2} \) |
| 41 | \( 1 + (-1.70e5 + 1.70e5i)T - 1.94e11iT^{2} \) |
| 43 | \( 1 + (6.31e5 + 2.61e5i)T + (1.92e11 + 1.92e11i)T^{2} \) |
| 47 | \( 1 + 3.84e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 + (-7.90e5 - 3.27e5i)T + (8.30e11 + 8.30e11i)T^{2} \) |
| 59 | \( 1 + (-7.43e5 + 1.79e6i)T + (-1.75e12 - 1.75e12i)T^{2} \) |
| 61 | \( 1 + (2.02e6 - 8.37e5i)T + (2.22e12 - 2.22e12i)T^{2} \) |
| 67 | \( 1 + (7.99e5 - 3.31e5i)T + (4.28e12 - 4.28e12i)T^{2} \) |
| 71 | \( 1 + (2.25e6 + 2.25e6i)T + 9.09e12iT^{2} \) |
| 73 | \( 1 + (3.58e6 - 3.58e6i)T - 1.10e13iT^{2} \) |
| 79 | \( 1 + 5.44e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 + (-2.73e6 - 6.60e6i)T + (-1.91e13 + 1.91e13i)T^{2} \) |
| 89 | \( 1 + (-4.81e6 - 4.81e6i)T + 4.42e13iT^{2} \) |
| 97 | \( 1 - 1.22e7T + 8.07e13T^{2} \) |
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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
−15.20660610223599384318142692680, −13.36122087240058641383065799265, −11.93622258979258481714419656666, −10.97644267210144655249037506369, −10.28016360536936718451483509356, −8.551136119209880668016928084411, −6.89186869349204665585146820109, −4.84621007261836449560392446663, −2.85152213748214701036779606851, −0.33987221490162734746476170388,
1.13582409961588461092006719587, 4.91092415192278433245582374079, 6.16088203666146891514754596465, 7.55600432662113054027592050670, 8.937788676960158453739143783129, 10.49989665634058444749658259789, 11.78927598386521012169491180019, 13.07868126241532369248088695262, 14.68832830531997393720022091627, 15.97140745188721183414452094685
