| L(s) = 1 | + (−4.50 + 6.61i)2-s + (11.0 − 11.0i)3-s + (−23.4 − 59.5i)4-s + (−9.26 + 9.26i)5-s + (23.2 + 122. i)6-s − 320.·7-s + (499. + 112. i)8-s − 242. i·9-s + (−19.5 − 103. i)10-s + (1.49e3 + 1.49e3i)11-s + (−914. − 397. i)12-s + (2.54e3 + 2.54e3i)13-s + (1.44e3 − 2.11e3i)14-s + 204. i·15-s + (−2.99e3 + 2.79e3i)16-s − 7.23e3·17-s + ⋯ |
| L(s) = 1 | + (−0.562 + 0.826i)2-s + (0.408 − 0.408i)3-s + (−0.366 − 0.930i)4-s + (−0.0741 + 0.0741i)5-s + (0.107 + 0.567i)6-s − 0.934·7-s + (0.975 + 0.220i)8-s − 0.333i·9-s + (−0.0195 − 0.103i)10-s + (1.12 + 1.12i)11-s + (−0.529 − 0.230i)12-s + (1.15 + 1.15i)13-s + (0.525 − 0.772i)14-s + 0.0605i·15-s + (−0.731 + 0.682i)16-s − 1.47·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.350 - 0.936i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.350 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.635260 + 0.915860i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.635260 + 0.915860i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (4.50 - 6.61i)T \) |
| 3 | \( 1 + (-11.0 + 11.0i)T \) |
| good | 5 | \( 1 + (9.26 - 9.26i)T - 1.56e4iT^{2} \) |
| 7 | \( 1 + 320.T + 1.17e5T^{2} \) |
| 11 | \( 1 + (-1.49e3 - 1.49e3i)T + 1.77e6iT^{2} \) |
| 13 | \( 1 + (-2.54e3 - 2.54e3i)T + 4.82e6iT^{2} \) |
| 17 | \( 1 + 7.23e3T + 2.41e7T^{2} \) |
| 19 | \( 1 + (4.86e3 - 4.86e3i)T - 4.70e7iT^{2} \) |
| 23 | \( 1 - 1.22e4T + 1.48e8T^{2} \) |
| 29 | \( 1 + (-9.52e3 - 9.52e3i)T + 5.94e8iT^{2} \) |
| 31 | \( 1 - 1.35e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + (964. - 964. i)T - 2.56e9iT^{2} \) |
| 41 | \( 1 + 9.46e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + (-6.14e4 - 6.14e4i)T + 6.32e9iT^{2} \) |
| 47 | \( 1 - 1.25e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 + (-9.27e4 + 9.27e4i)T - 2.21e10iT^{2} \) |
| 59 | \( 1 + (7.89e4 + 7.89e4i)T + 4.21e10iT^{2} \) |
| 61 | \( 1 + (-1.40e4 - 1.40e4i)T + 5.15e10iT^{2} \) |
| 67 | \( 1 + (1.69e5 - 1.69e5i)T - 9.04e10iT^{2} \) |
| 71 | \( 1 - 2.39e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + 6.68e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 - 5.72e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + (3.60e5 - 3.60e5i)T - 3.26e11iT^{2} \) |
| 89 | \( 1 - 1.01e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 1.29e6T + 8.32e11T^{2} \) |
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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
−14.87812022835028875063571587999, −13.82578106152467092674876644662, −12.71693819010871780580629801465, −11.03485165835466561375944385729, −9.414770931688585603739472740918, −8.799293224283402680712141444447, −6.99143606372750268598805660783, −6.43398765749147827663945744536, −4.15370546942672452512336690908, −1.59775341222094066136884341809,
0.61443094347938687332655214388, 2.88064342159563408062237284056, 4.06075174433573952450096491377, 6.47021615119199657095307693450, 8.471863511405099432112325547670, 9.092862292419090076456462218652, 10.50673759872512601423563694528, 11.38393095461266609925280501712, 12.97688947855031912281982934073, 13.63202054701473441677907134495
