| L(s) = 1 | + (7.81 + 5.67i)2-s + (2.78 − 8.55i)3-s + (18.9 + 58.2i)4-s + (25.1 − 18.2i)5-s + (70.2 − 51.0i)6-s + (56.7 + 174. i)7-s + (−87.1 + 268. i)8-s + (−65.5 − 47.6i)9-s + 299.·10-s + (−30.5 − 400. i)11-s + 550.·12-s + (−707. − 513. i)13-s + (−547. + 1.68e3i)14-s + (−86.4 − 265. i)15-s + (−617. + 448. i)16-s + (−627. + 455. i)17-s + ⋯ |
| L(s) = 1 | + (1.38 + 1.00i)2-s + (0.178 − 0.549i)3-s + (0.591 + 1.81i)4-s + (0.449 − 0.326i)5-s + (0.797 − 0.579i)6-s + (0.437 + 1.34i)7-s + (−0.481 + 1.48i)8-s + (−0.269 − 0.195i)9-s + 0.948·10-s + (−0.0761 − 0.997i)11-s + 1.10·12-s + (−1.16 − 0.843i)13-s + (−0.746 + 2.29i)14-s + (−0.0991 − 0.305i)15-s + (−0.603 + 0.438i)16-s + (−0.526 + 0.382i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.504 - 0.863i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.504 - 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.88869 + 1.65864i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.88869 + 1.65864i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.78 + 8.55i)T \) |
| 11 | \( 1 + (30.5 + 400. i)T \) |
| good | 2 | \( 1 + (-7.81 - 5.67i)T + (9.88 + 30.4i)T^{2} \) |
| 5 | \( 1 + (-25.1 + 18.2i)T + (965. - 2.97e3i)T^{2} \) |
| 7 | \( 1 + (-56.7 - 174. i)T + (-1.35e4 + 9.87e3i)T^{2} \) |
| 13 | \( 1 + (707. + 513. i)T + (1.14e5 + 3.53e5i)T^{2} \) |
| 17 | \( 1 + (627. - 455. i)T + (4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-462. + 1.42e3i)T + (-2.00e6 - 1.45e6i)T^{2} \) |
| 23 | \( 1 - 1.30e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + (150. + 461. i)T + (-1.65e7 + 1.20e7i)T^{2} \) |
| 31 | \( 1 + (6.11e3 + 4.43e3i)T + (8.84e6 + 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-3.61e3 - 1.11e4i)T + (-5.61e7 + 4.07e7i)T^{2} \) |
| 41 | \( 1 + (-2.55e3 + 7.86e3i)T + (-9.37e7 - 6.80e7i)T^{2} \) |
| 43 | \( 1 + 1.74e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (8.23e3 - 2.53e4i)T + (-1.85e8 - 1.34e8i)T^{2} \) |
| 53 | \( 1 + (-1.94e4 - 1.41e4i)T + (1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (-1.05e4 - 3.23e4i)T + (-5.78e8 + 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-3.14e4 + 2.28e4i)T + (2.60e8 - 8.03e8i)T^{2} \) |
| 67 | \( 1 + 7.05e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (-8.87e3 + 6.45e3i)T + (5.57e8 - 1.71e9i)T^{2} \) |
| 73 | \( 1 + (1.11e4 + 3.42e4i)T + (-1.67e9 + 1.21e9i)T^{2} \) |
| 79 | \( 1 + (-9.59e3 - 6.97e3i)T + (9.50e8 + 2.92e9i)T^{2} \) |
| 83 | \( 1 + (-7.78e3 + 5.65e3i)T + (1.21e9 - 3.74e9i)T^{2} \) |
| 89 | \( 1 - 9.33e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (6.63e4 + 4.81e4i)T + (2.65e9 + 8.16e9i)T^{2} \) |
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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
−15.39926196297326636419921499538, −14.78167749991673407123931628951, −13.46564939125180668017508807081, −12.74111202967739095880669312467, −11.57414081050283686409880494060, −8.946060837455970510783978402909, −7.61415227049054631893632421048, −5.98853844050931573520878900898, −5.12419498423404580102127315384, −2.75169640285549366514345445626,
2.08702741722323501567601130451, 3.96768585006325106771356343535, 5.00500919765915392047468665172, 7.09840190864753037348039730767, 9.803872025030403612691500167098, 10.60716822515547379462238205870, 11.82630647228489629022960442452, 13.16589713121952722782843539781, 14.33306288307847027899240319768, 14.66570290303420608038795087705
