| L(s) = 1 | + (0.826 − 0.563i)2-s + (−1.83 + 1.70i)3-s + (0.365 − 0.930i)4-s + (−2.84 − 0.878i)5-s + (−0.557 + 2.44i)6-s + (−0.222 − 0.974i)8-s + (0.245 − 3.27i)9-s + (−2.84 + 0.878i)10-s + (0.104 + 1.40i)11-s + (0.916 + 2.33i)12-s + (−0.0398 − 0.0191i)13-s + (6.73 − 3.24i)15-s + (−0.733 − 0.680i)16-s + (7.07 + 1.06i)17-s + (−1.64 − 2.84i)18-s + (3.92 − 6.79i)19-s + ⋯ |
| L(s) = 1 | + (0.584 − 0.398i)2-s + (−1.06 + 0.984i)3-s + (0.182 − 0.465i)4-s + (−1.27 − 0.393i)5-s + (−0.227 + 0.997i)6-s + (−0.0786 − 0.344i)8-s + (0.0818 − 1.09i)9-s + (−0.901 + 0.277i)10-s + (0.0316 + 0.422i)11-s + (0.264 + 0.673i)12-s + (−0.0110 − 0.00532i)13-s + (1.73 − 0.837i)15-s + (−0.183 − 0.170i)16-s + (1.71 + 0.258i)17-s + (−0.387 − 0.671i)18-s + (0.899 − 1.55i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 686 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 + 0.389i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 686 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.921 + 0.389i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.11563 - 0.226055i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.11563 - 0.226055i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.826 + 0.563i)T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + (1.83 - 1.70i)T + (0.224 - 2.99i)T^{2} \) |
| 5 | \( 1 + (2.84 + 0.878i)T + (4.13 + 2.81i)T^{2} \) |
| 11 | \( 1 + (-0.104 - 1.40i)T + (-10.8 + 1.63i)T^{2} \) |
| 13 | \( 1 + (0.0398 + 0.0191i)T + (8.10 + 10.1i)T^{2} \) |
| 17 | \( 1 + (-7.07 - 1.06i)T + (16.2 + 5.01i)T^{2} \) |
| 19 | \( 1 + (-3.92 + 6.79i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.74 + 0.865i)T + (21.9 - 6.77i)T^{2} \) |
| 29 | \( 1 + (2.49 - 3.12i)T + (-6.45 - 28.2i)T^{2} \) |
| 31 | \( 1 + (-2.20 - 3.82i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.299 - 0.764i)T + (-27.1 + 25.1i)T^{2} \) |
| 41 | \( 1 + (-0.416 - 1.82i)T + (-36.9 + 17.7i)T^{2} \) |
| 43 | \( 1 + (-0.802 + 3.51i)T + (-38.7 - 18.6i)T^{2} \) |
| 47 | \( 1 + (-4.07 + 2.77i)T + (17.1 - 43.7i)T^{2} \) |
| 53 | \( 1 + (-2.07 + 5.28i)T + (-38.8 - 36.0i)T^{2} \) |
| 59 | \( 1 + (-13.3 + 4.11i)T + (48.7 - 33.2i)T^{2} \) |
| 61 | \( 1 + (-0.856 - 2.18i)T + (-44.7 + 41.4i)T^{2} \) |
| 67 | \( 1 + (6.80 + 11.7i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.01 - 2.52i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (4.74 + 3.23i)T + (26.6 + 67.9i)T^{2} \) |
| 79 | \( 1 + (-4.13 + 7.16i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.263 + 0.126i)T + (51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (0.794 - 10.6i)T + (-88.0 - 13.2i)T^{2} \) |
| 97 | \( 1 + 12.1T + 97T^{2} \) |
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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
−10.66368796328052597549262250531, −9.848834037874293231503092871291, −8.933510519487823767442605640680, −7.67066518431304785961342368650, −6.77777627256645178388059943937, −5.32598580489013603733440299517, −5.00526403013375417539306688471, −4.02498519969128934820483394076, −3.16527596055893715085757744787, −0.791604139384104662062029485590,
1.02619154745761541638218937103, 3.10542304068500837604777868658, 4.03193240272122587614171550660, 5.48002183194055806142098449820, 5.90953912814062836395896031354, 7.18132267534474478527285259324, 7.50334369411794559189820340783, 8.269688149972676748299017753130, 9.832841622903837752500900306056, 11.00971669869405244562848266260
