| L(s) = 1 | + (−2.12 − 0.772i)2-s + (−0.789 + 5.13i)3-s + (−2.22 − 1.86i)4-s + (−3.33 + 18.9i)5-s + (5.64 − 10.2i)6-s + (0.500 − 0.420i)7-s + (12.3 + 21.3i)8-s + (−25.7 − 8.11i)9-s + (21.7 − 37.6i)10-s + (−4.24 − 24.0i)11-s + (11.3 − 9.94i)12-s + (36.6 − 13.3i)13-s + (−1.38 + 0.504i)14-s + (−94.6 − 32.1i)15-s + (−5.61 − 31.8i)16-s + (−20.0 + 34.6i)17-s + ⋯ |
| L(s) = 1 | + (−0.750 − 0.273i)2-s + (−0.151 + 0.988i)3-s + (−0.277 − 0.233i)4-s + (−0.298 + 1.69i)5-s + (0.383 − 0.699i)6-s + (0.0270 − 0.0226i)7-s + (0.543 + 0.942i)8-s + (−0.953 − 0.300i)9-s + (0.686 − 1.18i)10-s + (−0.116 − 0.659i)11-s + (0.272 − 0.239i)12-s + (0.781 − 0.284i)13-s + (−0.0264 + 0.00963i)14-s + (−1.62 − 0.552i)15-s + (−0.0878 − 0.497i)16-s + (−0.285 + 0.494i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.294 - 0.955i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.294 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.354687 + 0.480321i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.354687 + 0.480321i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.789 - 5.13i)T \) |
| good | 2 | \( 1 + (2.12 + 0.772i)T + (6.12 + 5.14i)T^{2} \) |
| 5 | \( 1 + (3.33 - 18.9i)T + (-117. - 42.7i)T^{2} \) |
| 7 | \( 1 + (-0.500 + 0.420i)T + (59.5 - 337. i)T^{2} \) |
| 11 | \( 1 + (4.24 + 24.0i)T + (-1.25e3 + 455. i)T^{2} \) |
| 13 | \( 1 + (-36.6 + 13.3i)T + (1.68e3 - 1.41e3i)T^{2} \) |
| 17 | \( 1 + (20.0 - 34.6i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-76.3 - 132. i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-89.8 - 75.4i)T + (2.11e3 + 1.19e4i)T^{2} \) |
| 29 | \( 1 + (83.5 + 30.4i)T + (1.86e4 + 1.56e4i)T^{2} \) |
| 31 | \( 1 + (-58.9 - 49.4i)T + (5.17e3 + 2.93e4i)T^{2} \) |
| 37 | \( 1 + (-108. + 187. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (48.1 - 17.5i)T + (5.27e4 - 4.43e4i)T^{2} \) |
| 43 | \( 1 + (-26.9 - 152. i)T + (-7.47e4 + 2.71e4i)T^{2} \) |
| 47 | \( 1 + (-100. + 84.2i)T + (1.80e4 - 1.02e5i)T^{2} \) |
| 53 | \( 1 + 362.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-60.3 + 342. i)T + (-1.92e5 - 7.02e4i)T^{2} \) |
| 61 | \( 1 + (-279. + 234. i)T + (3.94e4 - 2.23e5i)T^{2} \) |
| 67 | \( 1 + (-749. + 272. i)T + (2.30e5 - 1.93e5i)T^{2} \) |
| 71 | \( 1 + (185. - 320. i)T + (-1.78e5 - 3.09e5i)T^{2} \) |
| 73 | \( 1 + (253. + 438. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (143. + 52.1i)T + (3.77e5 + 3.16e5i)T^{2} \) |
| 83 | \( 1 + (-457. - 166. i)T + (4.38e5 + 3.67e5i)T^{2} \) |
| 89 | \( 1 + (-44.9 - 77.8i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-197. - 1.11e3i)T + (-8.57e5 + 3.12e5i)T^{2} \) |
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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
−17.45464229534682582317229265412, −15.98625603577678809018667325972, −14.79396146210322405645640898399, −13.96404769714766129064834373138, −11.27211842028167878124929406706, −10.72475569097974574075323535154, −9.645973874038122501382679479620, −8.042980729264788706290229597012, −5.86901509252705996686486337698, −3.47578625177319230596792451407,
0.825891913404535097636267097145, 4.84347307980598724335577087435, 7.14432603841726370368899543670, 8.438186392234704453265647420931, 9.242262122640349428930311305613, 11.62121803666297794156862900784, 12.87348505311779030933913625672, 13.45806776849141388411865343839, 15.79398171338706201525471537378, 16.80132581173430601254384319917
