| L(s) = 1 | + (−1.40 − 2.65i)3-s + (1.05 − 0.786i)5-s + (−3.63 + 2.39i)7-s + (−5.07 + 7.43i)9-s + (0.102 − 0.879i)11-s + (−5.05 − 16.8i)13-s + (−3.56 − 1.70i)15-s + (−25.3 − 4.47i)17-s + (1.99 + 11.3i)19-s + (11.4 + 6.29i)21-s + (−22.2 + 33.7i)23-s + (−6.67 + 22.2i)25-s + (26.8 + 3.04i)27-s + (26.3 − 24.8i)29-s + (−2.94 + 50.5i)31-s + ⋯ |
| L(s) = 1 | + (−0.466 − 0.884i)3-s + (0.211 − 0.157i)5-s + (−0.519 + 0.341i)7-s + (−0.563 + 0.825i)9-s + (0.00934 − 0.0799i)11-s + (−0.388 − 1.29i)13-s + (−0.237 − 0.113i)15-s + (−1.49 − 0.262i)17-s + (0.105 + 0.596i)19-s + (0.545 + 0.299i)21-s + (−0.965 + 1.46i)23-s + (−0.266 + 0.891i)25-s + (0.993 + 0.112i)27-s + (0.908 − 0.856i)29-s + (−0.0949 + 1.62i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.756 - 0.654i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.756 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.00462015 + 0.0124008i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00462015 + 0.0124008i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.40 + 2.65i)T \) |
| good | 5 | \( 1 + (-1.05 + 0.786i)T + (7.17 - 23.9i)T^{2} \) |
| 7 | \( 1 + (3.63 - 2.39i)T + (19.4 - 44.9i)T^{2} \) |
| 11 | \( 1 + (-0.102 + 0.879i)T + (-117. - 27.9i)T^{2} \) |
| 13 | \( 1 + (5.05 + 16.8i)T + (-141. + 92.8i)T^{2} \) |
| 17 | \( 1 + (25.3 + 4.47i)T + (271. + 98.8i)T^{2} \) |
| 19 | \( 1 + (-1.99 - 11.3i)T + (-339. + 123. i)T^{2} \) |
| 23 | \( 1 + (22.2 - 33.7i)T + (-209. - 485. i)T^{2} \) |
| 29 | \( 1 + (-26.3 + 24.8i)T + (48.8 - 839. i)T^{2} \) |
| 31 | \( 1 + (2.94 - 50.5i)T + (-954. - 111. i)T^{2} \) |
| 37 | \( 1 + (52.9 - 19.2i)T + (1.04e3 - 879. i)T^{2} \) |
| 41 | \( 1 + (-10.9 + 46.3i)T + (-1.50e3 - 754. i)T^{2} \) |
| 43 | \( 1 + (16.1 + 37.3i)T + (-1.26e3 + 1.34e3i)T^{2} \) |
| 47 | \( 1 + (11.3 - 0.658i)T + (2.19e3 - 256. i)T^{2} \) |
| 53 | \( 1 + (19.9 + 11.5i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (1.62 + 13.9i)T + (-3.38e3 + 802. i)T^{2} \) |
| 61 | \( 1 + (41.2 + 20.7i)T + (2.22e3 + 2.98e3i)T^{2} \) |
| 67 | \( 1 + (42.5 - 45.0i)T + (-261. - 4.48e3i)T^{2} \) |
| 71 | \( 1 + (-80.7 + 96.2i)T + (-875. - 4.96e3i)T^{2} \) |
| 73 | \( 1 + (-59.5 + 49.9i)T + (925. - 5.24e3i)T^{2} \) |
| 79 | \( 1 + (35.6 - 8.45i)T + (5.57e3 - 2.80e3i)T^{2} \) |
| 83 | \( 1 + (1.52 + 6.45i)T + (-6.15e3 + 3.09e3i)T^{2} \) |
| 89 | \( 1 + (73.1 + 87.1i)T + (-1.37e3 + 7.80e3i)T^{2} \) |
| 97 | \( 1 + (59.2 - 79.5i)T + (-2.69e3 - 9.01e3i)T^{2} \) |
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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
−10.89738153647727102771674462821, −9.993714186791017855829571410176, −8.834419964574133375929130294311, −7.86102907179923564535627587339, −6.88932302476186215634410361706, −5.88948333534344398681316674056, −5.09082123374979315108811918576, −3.23055208539178179368569007148, −1.82160784102673821395450216072, −0.00617348721117299468539838756,
2.44826548013662517440067745335, 4.07130829041737332011791404746, 4.72358465213000308423540472223, 6.31673956111346160255662517738, 6.73781046601239196808719068643, 8.456681575021238093208583891392, 9.364045901276588963159984003121, 10.11162673832729944014337828345, 10.95702267949497562967328256168, 11.77554290110940446376033902281
