| L(s) = 1 | + (−9.69 − 3.52i)2-s + (−5.72 + 14.4i)3-s + (57.0 + 47.8i)4-s + (2.39 − 13.6i)5-s + (106. − 120. i)6-s + (−83.8 + 70.3i)7-s + (−218. − 379. i)8-s + (−177. − 166. i)9-s + (−71.2 + 123. i)10-s + (−96.8 − 549. i)11-s + (−1.02e3 + 552. i)12-s + (857. − 312. i)13-s + (1.06e3 − 386. i)14-s + (183. + 112. i)15-s + (370. + 2.10e3i)16-s + (−249. + 431. i)17-s + ⋯ |
| L(s) = 1 | + (−1.71 − 0.623i)2-s + (−0.367 + 0.930i)3-s + (1.78 + 1.49i)4-s + (0.0429 − 0.243i)5-s + (1.20 − 1.36i)6-s + (−0.647 + 0.542i)7-s + (−1.20 − 2.09i)8-s + (−0.730 − 0.683i)9-s + (−0.225 + 0.390i)10-s + (−0.241 − 1.36i)11-s + (−2.04 + 1.10i)12-s + (1.40 − 0.512i)13-s + (1.44 − 0.526i)14-s + (0.210 + 0.129i)15-s + (0.362 + 2.05i)16-s + (−0.209 + 0.362i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.363 + 0.931i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.363 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.173059 - 0.253304i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.173059 - 0.253304i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (5.72 - 14.4i)T \) |
| good | 2 | \( 1 + (9.69 + 3.52i)T + (24.5 + 20.5i)T^{2} \) |
| 5 | \( 1 + (-2.39 + 13.6i)T + (-2.93e3 - 1.06e3i)T^{2} \) |
| 7 | \( 1 + (83.8 - 70.3i)T + (2.91e3 - 1.65e4i)T^{2} \) |
| 11 | \( 1 + (96.8 + 549. i)T + (-1.51e5 + 5.50e4i)T^{2} \) |
| 13 | \( 1 + (-857. + 312. i)T + (2.84e5 - 2.38e5i)T^{2} \) |
| 17 | \( 1 + (249. - 431. i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 19 | \( 1 + (1.43e3 + 2.47e3i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (296. + 249. i)T + (1.11e6 + 6.33e6i)T^{2} \) |
| 29 | \( 1 + (612. + 222. i)T + (1.57e7 + 1.31e7i)T^{2} \) |
| 31 | \( 1 + (4.64e3 + 3.89e3i)T + (4.97e6 + 2.81e7i)T^{2} \) |
| 37 | \( 1 + (-325. + 563. i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 + (-3.37e3 + 1.22e3i)T + (8.87e7 - 7.44e7i)T^{2} \) |
| 43 | \( 1 + (2.55e3 + 1.44e4i)T + (-1.38e8 + 5.02e7i)T^{2} \) |
| 47 | \( 1 + (1.17e4 - 9.85e3i)T + (3.98e7 - 2.25e8i)T^{2} \) |
| 53 | \( 1 + 2.30e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + (-275. + 1.56e3i)T + (-6.71e8 - 2.44e8i)T^{2} \) |
| 61 | \( 1 + (2.31e3 - 1.94e3i)T + (1.46e8 - 8.31e8i)T^{2} \) |
| 67 | \( 1 + (6.27e3 - 2.28e3i)T + (1.03e9 - 8.67e8i)T^{2} \) |
| 71 | \( 1 + (-1.78e4 + 3.09e4i)T + (-9.02e8 - 1.56e9i)T^{2} \) |
| 73 | \( 1 + (2.32e4 + 4.03e4i)T + (-1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (1.38e4 + 5.04e3i)T + (2.35e9 + 1.97e9i)T^{2} \) |
| 83 | \( 1 + (6.88e4 + 2.50e4i)T + (3.01e9 + 2.53e9i)T^{2} \) |
| 89 | \( 1 + (1.70e4 + 2.94e4i)T + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 + (2.40e3 + 1.36e4i)T + (-8.06e9 + 2.93e9i)T^{2} \) |
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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
−16.22339168204876822741817947810, −15.51444898388797411327320004297, −12.94250161588326111800478619137, −11.23676761747531666142267971250, −10.71415002126268946418234190593, −9.149273879428941154670263916873, −8.565399607990870991794505113325, −6.16520749361222063815554699036, −3.16582412336962924068589087086, −0.36689057539596026537878849194,
1.59984733964207088328073344245, 6.29008981487749903760322123526, 7.13501780986485790773824966343, 8.423000526501111399321740488111, 9.995208247883543688377127787651, 11.07650774151586278090228824256, 12.77083302299176946161784320490, 14.50210773078068208369064769656, 16.08836929225240419894249005066, 16.84810437038968786900109073514
