| L(s) = 1 | + (−2.65 − 8.17i)2-s + (7.28 + 5.29i)3-s + (−33.8 + 24.5i)4-s + (−21.1 + 65.1i)5-s + (23.8 − 73.5i)6-s + (−196. + 142. i)7-s + (68.1 + 49.5i)8-s + (25.0 + 77.0i)9-s + 588.·10-s + (375. + 141. i)11-s − 376.·12-s + (−286. − 881. i)13-s + (1.68e3 + 1.22e3i)14-s + (−499. + 362. i)15-s + (−189. + 583. i)16-s + (−344. + 1.06e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.469 − 1.44i)2-s + (0.467 + 0.339i)3-s + (−1.05 + 0.767i)4-s + (−0.378 + 1.16i)5-s + (0.270 − 0.833i)6-s + (−1.51 + 1.10i)7-s + (0.376 + 0.273i)8-s + (0.103 + 0.317i)9-s + 1.86·10-s + (0.935 + 0.352i)11-s − 0.754·12-s + (−0.470 − 1.44i)13-s + (2.30 + 1.67i)14-s + (−0.572 + 0.416i)15-s + (−0.185 + 0.570i)16-s + (−0.289 + 0.890i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.395 - 0.918i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.395 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.521978 + 0.343732i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.521978 + 0.343732i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-7.28 - 5.29i)T \) |
| 11 | \( 1 + (-375. - 141. i)T \) |
| good | 2 | \( 1 + (2.65 + 8.17i)T + (-25.8 + 18.8i)T^{2} \) |
| 5 | \( 1 + (21.1 - 65.1i)T + (-2.52e3 - 1.83e3i)T^{2} \) |
| 7 | \( 1 + (196. - 142. i)T + (5.19e3 - 1.59e4i)T^{2} \) |
| 13 | \( 1 + (286. + 881. i)T + (-3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (344. - 1.06e3i)T + (-1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (1.00e3 + 730. i)T + (7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 - 40.4T + 6.43e6T^{2} \) |
| 29 | \( 1 + (-439. + 319. i)T + (6.33e6 - 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-1.93e3 - 5.95e3i)T + (-2.31e7 + 1.68e7i)T^{2} \) |
| 37 | \( 1 + (7.45e3 - 5.41e3i)T + (2.14e7 - 6.59e7i)T^{2} \) |
| 41 | \( 1 + (-6.96e3 - 5.05e3i)T + (3.58e7 + 1.10e8i)T^{2} \) |
| 43 | \( 1 + 2.23e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-550. - 399. i)T + (7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-1.60e3 - 4.94e3i)T + (-3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (-9.70e3 + 7.05e3i)T + (2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 + (741. - 2.28e3i)T + (-6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 + 9.77e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + (-2.35e3 + 7.25e3i)T + (-1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (2.29e4 - 1.66e4i)T + (6.40e8 - 1.97e9i)T^{2} \) |
| 79 | \( 1 + (-2.18e4 - 6.72e4i)T + (-2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (-7.34e3 + 2.26e4i)T + (-3.18e9 - 2.31e9i)T^{2} \) |
| 89 | \( 1 - 1.54e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (-1.78e4 - 5.48e4i)T + (-6.94e9 + 5.04e9i)T^{2} \) |
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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
−15.56960501921009365781804804667, −14.91231106493599233329186284967, −12.97222379064399905086091217896, −12.11524321307979065365619813825, −10.66853880078072738288698281745, −9.851824698681109885109634988057, −8.702052980305582514508608699027, −6.53373402981098854981838433219, −3.47833764767195475678844035804, −2.63210180369337260214909312380,
0.40512653750120835846405174456, 4.19605611821176497197516891447, 6.42530410361358046914611006767, 7.31654639485901418030944177240, 8.844782843680215231839046278459, 9.534391407275473012224449247525, 12.06827445820480888074388625726, 13.42993969939209778099053789927, 14.34930461845827439975658190903, 15.95925640023896181387589592460
