| L(s) = 1 | + (−0.969 + 1.02i)2-s + (0.755 − 0.654i)3-s + (−0.121 − 1.99i)4-s + (−2.07 + 0.298i)5-s + (−0.0579 + 1.41i)6-s + (1.51 − 3.31i)7-s + (2.17 + 1.80i)8-s + (0.142 − 0.989i)9-s + (1.70 − 2.42i)10-s + (−1.05 + 3.58i)11-s + (−1.39 − 1.42i)12-s + (1.79 − 0.821i)13-s + (1.94 + 4.77i)14-s + (−1.37 + 1.58i)15-s + (−3.97 + 0.485i)16-s + (−5.87 − 3.77i)17-s + ⋯ |
| L(s) = 1 | + (−0.685 + 0.728i)2-s + (0.436 − 0.378i)3-s + (−0.0608 − 0.998i)4-s + (−0.928 + 0.133i)5-s + (−0.0236 + 0.576i)6-s + (0.572 − 1.25i)7-s + (0.768 + 0.639i)8-s + (0.0474 − 0.329i)9-s + (0.539 − 0.768i)10-s + (−0.317 + 1.08i)11-s + (−0.403 − 0.412i)12-s + (0.498 − 0.227i)13-s + (0.520 + 1.27i)14-s + (−0.354 + 0.409i)15-s + (−0.992 + 0.121i)16-s + (−1.42 − 0.915i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0773 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0773 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.541773 - 0.501371i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.541773 - 0.501371i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.969 - 1.02i)T \) |
| 3 | \( 1 + (-0.755 + 0.654i)T \) |
| 23 | \( 1 + (1.06 + 4.67i)T \) |
| good | 5 | \( 1 + (2.07 - 0.298i)T + (4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (-1.51 + 3.31i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (1.05 - 3.58i)T + (-9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (-1.79 + 0.821i)T + (8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (5.87 + 3.77i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (0.728 + 1.13i)T + (-7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-4.71 + 7.34i)T + (-12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-2.50 + 2.89i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (6.24 + 0.897i)T + (35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (0.777 + 5.40i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (3.65 - 3.16i)T + (6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + 4.13T + 47T^{2} \) |
| 53 | \( 1 + (-6.04 - 2.76i)T + (34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (3.91 - 1.78i)T + (38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (-3.49 - 3.02i)T + (8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (1.71 + 5.84i)T + (-56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (-8.60 + 2.52i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (-12.0 + 7.75i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (-0.356 - 0.779i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (9.02 + 1.29i)T + (79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-9.92 - 11.4i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (-1.57 - 10.9i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−10.51436948997409753596573168538, −9.595137338254850491795878991466, −8.459741437707411478462566373623, −7.86996549012883003691890017477, −7.14833489531785092065545683426, −6.54026178134355028070956897119, −4.77890848192339335644898606361, −4.13571297923186465379923753931, −2.20591669000533743494797856303, −0.50298260928968874269450999258,
1.80503406704238814908619608504, 3.10718207488193421677234464339, 3.99298231307158248188326861770, 5.17973467124569134045673535857, 6.63655843717561037238170124439, 8.075450737480566797484961503833, 8.520448497078072518773683926780, 8.899182893648663200580264453385, 10.19951795020541127798969876734, 11.12210353618618024895009542979
