| L(s) = 1 | + (7.22 − 5.76i)2-s + (7.33 − 48.6i)3-s + (4.77 − 20.9i)4-s + (108. − 158. i)5-s + (−227. − 394. i)6-s + (411. + 237. i)7-s + (170. + 354. i)8-s + (−1.61e3 − 499. i)9-s + (−132. − 1.77e3i)10-s + (61.7 + 270. i)11-s + (−984. − 386. i)12-s + (−159. + 2.13e3i)13-s + (4.34e3 − 654. i)14-s + (−6.93e3 − 6.43e3i)15-s + (4.51e3 + 2.17e3i)16-s + (−5.87e3 + 4.00e3i)17-s + ⋯ |
| L(s) = 1 | + (0.903 − 0.720i)2-s + (0.271 − 1.80i)3-s + (0.0746 − 0.327i)4-s + (0.866 − 1.27i)5-s + (−1.05 − 1.82i)6-s + (1.20 + 0.692i)7-s + (0.333 + 0.691i)8-s + (−2.22 − 0.685i)9-s + (−0.132 − 1.77i)10-s + (0.0464 + 0.203i)11-s + (−0.569 − 0.223i)12-s + (−0.0727 + 0.970i)13-s + (1.58 − 0.238i)14-s + (−2.05 − 1.90i)15-s + (1.10 + 0.530i)16-s + (−1.19 + 0.815i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.748 + 0.662i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.748 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.23922 - 3.26953i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.23922 - 3.26953i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-2.83e4 - 7.42e4i)T \) |
| good | 2 | \( 1 + (-7.22 + 5.76i)T + (14.2 - 62.3i)T^{2} \) |
| 3 | \( 1 + (-7.33 + 48.6i)T + (-696. - 214. i)T^{2} \) |
| 5 | \( 1 + (-108. + 158. i)T + (-5.70e3 - 1.45e4i)T^{2} \) |
| 7 | \( 1 + (-411. - 237. i)T + (5.88e4 + 1.01e5i)T^{2} \) |
| 11 | \( 1 + (-61.7 - 270. i)T + (-1.59e6 + 7.68e5i)T^{2} \) |
| 13 | \( 1 + (159. - 2.13e3i)T + (-4.77e6 - 7.19e5i)T^{2} \) |
| 17 | \( 1 + (5.87e3 - 4.00e3i)T + (8.81e6 - 2.24e7i)T^{2} \) |
| 19 | \( 1 + (-1.74e3 - 5.64e3i)T + (-3.88e7 + 2.65e7i)T^{2} \) |
| 23 | \( 1 + (-6.13e3 + 5.69e3i)T + (1.10e7 - 1.47e8i)T^{2} \) |
| 29 | \( 1 + (4.21e3 + 2.79e4i)T + (-5.68e8 + 1.75e8i)T^{2} \) |
| 31 | \( 1 + (-3.34e3 + 8.51e3i)T + (-6.50e8 - 6.03e8i)T^{2} \) |
| 37 | \( 1 + (4.85e4 - 2.80e4i)T + (1.28e9 - 2.22e9i)T^{2} \) |
| 41 | \( 1 + (-6.16e3 - 7.73e3i)T + (-1.05e9 + 4.63e9i)T^{2} \) |
| 47 | \( 1 + (-1.06e4 + 4.67e4i)T + (-9.71e9 - 4.67e9i)T^{2} \) |
| 53 | \( 1 + (1.54e4 + 2.05e5i)T + (-2.19e10 + 3.30e9i)T^{2} \) |
| 59 | \( 1 + (3.68e5 + 1.77e5i)T + (2.62e10 + 3.29e10i)T^{2} \) |
| 61 | \( 1 + (7.69e4 - 3.02e4i)T + (3.77e10 - 3.50e10i)T^{2} \) |
| 67 | \( 1 + (6.20e4 - 1.91e4i)T + (7.47e10 - 5.09e10i)T^{2} \) |
| 71 | \( 1 + (1.83e3 - 1.98e3i)T + (-9.57e9 - 1.27e11i)T^{2} \) |
| 73 | \( 1 + (-2.21e4 - 1.66e3i)T + (1.49e11 + 2.25e10i)T^{2} \) |
| 79 | \( 1 + (-1.00e5 + 1.74e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + (2.98e5 + 4.50e4i)T + (3.12e11 + 9.63e10i)T^{2} \) |
| 89 | \( 1 + (-2.04e5 + 1.35e6i)T + (-4.74e11 - 1.46e11i)T^{2} \) |
| 97 | \( 1 + (-2.79e5 - 1.22e6i)T + (-7.50e11 + 3.61e11i)T^{2} \) |
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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
−13.74034882156868620668252057492, −13.00636341785153724969494113761, −12.21692050895545920112147790250, −11.45407735408827589037335161351, −8.851189298160124261337730865621, −8.060685678354341559904901242998, −6.09553343752862322275780227428, −4.76801403506932593896543193842, −2.07964045260721384452711934677, −1.62112879978280627118916559764,
3.09711357082945573738523313860, 4.59149076035575924676254042251, 5.52053441016048388403935666128, 7.20305458807208919148830567473, 9.215443796412210887231726957001, 10.59789040838207751603385496786, 10.89173469007603256693813589700, 13.76318975105060716207774111186, 14.14312679169372033944597369754, 15.11614125590759096457437287465
