| 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 | \( 1 + (-1.78e4 + 5.48e4i)T + (-6.94e9 - 5.04e9i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.95925640023896181387589592460, −14.34930461845827439975658190903, −13.42993969939209778099053789927, −12.06827445820480888074388625726, −9.534391407275473012224449247525, −8.844782843680215231839046278459, −7.31654639485901418030944177240, −6.42530410361358046914611006767, −4.19605611821176497197516891447, −0.40512653750120835846405174456,
2.63210180369337260214909312380, 3.47833764767195475678844035804, 6.53373402981098854981838433219, 8.702052980305582514508608699027, 9.851824698681109885109634988057, 10.66853880078072738288698281745, 12.11524321307979065365619813825, 12.97222379064399905086091217896, 14.91231106493599233329186284967, 15.56960501921009365781804804667
