| L(s) = 1 | + (−2.03 + 2.42i)2-s + (10.3 + 24.9i)3-s + (9.37 + 53.1i)4-s + (7.59 − 20.8i)5-s + (−81.4 − 25.6i)6-s + (0.247 − 1.40i)7-s + (−323. − 186. i)8-s + (−514. + 516. i)9-s + (35.1 + 60.8i)10-s + (116. + 319. i)11-s + (−1.22e3 + 784. i)12-s + (−1.82e3 + 1.52e3i)13-s + (2.90 + 3.45i)14-s + (599. − 26.6i)15-s + (−2.13e3 + 778. i)16-s + (2.68e3 − 1.54e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.254 + 0.302i)2-s + (0.383 + 0.923i)3-s + (0.146 + 0.830i)4-s + (0.0607 − 0.166i)5-s + (−0.377 − 0.118i)6-s + (0.000722 − 0.00409i)7-s + (−0.631 − 0.364i)8-s + (−0.706 + 0.708i)9-s + (0.0351 + 0.0608i)10-s + (0.0872 + 0.239i)11-s + (−0.711 + 0.453i)12-s + (−0.828 + 0.695i)13-s + (0.00105 + 0.00125i)14-s + (0.177 − 0.00788i)15-s + (−0.521 + 0.189i)16-s + (0.546 − 0.315i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.812 - 0.583i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.812 - 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.426006 + 1.32272i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.426006 + 1.32272i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-10.3 - 24.9i)T \) |
| good | 2 | \( 1 + (2.03 - 2.42i)T + (-11.1 - 63.0i)T^{2} \) |
| 5 | \( 1 + (-7.59 + 20.8i)T + (-1.19e4 - 1.00e4i)T^{2} \) |
| 7 | \( 1 + (-0.247 + 1.40i)T + (-1.10e5 - 4.02e4i)T^{2} \) |
| 11 | \( 1 + (-116. - 319. i)T + (-1.35e6 + 1.13e6i)T^{2} \) |
| 13 | \( 1 + (1.82e3 - 1.52e3i)T + (8.38e5 - 4.75e6i)T^{2} \) |
| 17 | \( 1 + (-2.68e3 + 1.54e3i)T + (1.20e7 - 2.09e7i)T^{2} \) |
| 19 | \( 1 + (-352. + 609. i)T + (-2.35e7 - 4.07e7i)T^{2} \) |
| 23 | \( 1 + (-1.96e4 + 3.46e3i)T + (1.39e8 - 5.06e7i)T^{2} \) |
| 29 | \( 1 + (-2.92e3 + 3.49e3i)T + (-1.03e8 - 5.85e8i)T^{2} \) |
| 31 | \( 1 + (-5.88e3 - 3.33e4i)T + (-8.33e8 + 3.03e8i)T^{2} \) |
| 37 | \( 1 + (-1.37e4 - 2.37e4i)T + (-1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 + (-3.45e4 - 4.11e4i)T + (-8.24e8 + 4.67e9i)T^{2} \) |
| 43 | \( 1 + (6.33e4 - 2.30e4i)T + (4.84e9 - 4.06e9i)T^{2} \) |
| 47 | \( 1 + (-3.17e4 - 5.59e3i)T + (1.01e10 + 3.68e9i)T^{2} \) |
| 53 | \( 1 - 3.89e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 + (-1.00e5 + 2.75e5i)T + (-3.23e10 - 2.71e10i)T^{2} \) |
| 61 | \( 1 + (-4.66e4 + 2.64e5i)T + (-4.84e10 - 1.76e10i)T^{2} \) |
| 67 | \( 1 + (-3.37e5 + 2.83e5i)T + (1.57e10 - 8.90e10i)T^{2} \) |
| 71 | \( 1 + (3.31e5 - 1.91e5i)T + (6.40e10 - 1.10e11i)T^{2} \) |
| 73 | \( 1 + (1.08e5 - 1.88e5i)T + (-7.56e10 - 1.31e11i)T^{2} \) |
| 79 | \( 1 + (5.04e5 + 4.23e5i)T + (4.22e10 + 2.39e11i)T^{2} \) |
| 83 | \( 1 + (6.29e5 - 7.49e5i)T + (-5.67e10 - 3.21e11i)T^{2} \) |
| 89 | \( 1 + (9.91e5 + 5.72e5i)T + (2.48e11 + 4.30e11i)T^{2} \) |
| 97 | \( 1 + (-6.72e5 + 2.44e5i)T + (6.38e11 - 5.35e11i)T^{2} \) |
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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
−16.61648466704438183766031262192, −15.45868359863205087383171747611, −14.32003421594453361762478137822, −12.72820089559766432664549833754, −11.33956996884977869071949635066, −9.657110908324533669277274917317, −8.619458569329200421258235970739, −7.11648240488116578685956469767, −4.77196726504044151452300800851, −3.01038631120432816712126449778,
0.847061755858730658632939861053, 2.64463072870943660129471544207, 5.66979224713263050264624635148, 7.19114334531732779698827675598, 8.835242123146754348078036964993, 10.27056695864945417757800148855, 11.67406335228489253652836086331, 12.98954328654671044539132536878, 14.37652477368034020674702822265, 15.14041044449139951200124626488
