L(s) = 1 | + 8.17i·2-s + (−40.5 − 70.1i)3-s + 189.·4-s − 920. i·5-s + (572. − 331. i)6-s − 1.09e3·7-s + 3.63e3i·8-s + (−3.26e3 + 5.68e3i)9-s + 7.51e3·10-s − 4.41e3i·11-s + (−7.67e3 − 1.32e4i)12-s − 2.57e4·13-s − 8.93e3i·14-s + (−6.45e4 + 3.73e4i)15-s + 1.87e4·16-s − 3.74e4i·17-s + ⋯ |
L(s) = 1 | + 0.510i·2-s + (−0.500 − 0.865i)3-s + 0.739·4-s − 1.47i·5-s + (0.441 − 0.255i)6-s − 0.455·7-s + 0.888i·8-s + (−0.498 + 0.867i)9-s + 0.751·10-s − 0.301i·11-s + (−0.370 − 0.639i)12-s − 0.902·13-s − 0.232i·14-s + (−1.27 + 0.737i)15-s + 0.285·16-s − 0.447i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.865 + 0.500i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.865 + 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(0.233203 - 0.868595i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.233203 - 0.868595i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (40.5 + 70.1i)T \) |
| 11 | \( 1 + 4.41e3iT \) |
good | 2 | \( 1 - 8.17iT - 256T^{2} \) |
| 5 | \( 1 + 920. iT - 3.90e5T^{2} \) |
| 7 | \( 1 + 1.09e3T + 5.76e6T^{2} \) |
| 13 | \( 1 + 2.57e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + 3.74e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 1.91e5T + 1.69e10T^{2} \) |
| 23 | \( 1 + 2.57e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 2.72e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 1.44e6T + 8.52e11T^{2} \) |
| 37 | \( 1 - 2.40e5T + 3.51e12T^{2} \) |
| 41 | \( 1 + 3.01e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 6.24e6T + 1.16e13T^{2} \) |
| 47 | \( 1 - 3.04e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 6.80e5iT - 6.22e13T^{2} \) |
| 59 | \( 1 + 1.81e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 1.16e7T + 1.91e14T^{2} \) |
| 67 | \( 1 + 1.58e7T + 4.06e14T^{2} \) |
| 71 | \( 1 + 4.27e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 1.96e7T + 8.06e14T^{2} \) |
| 79 | \( 1 - 4.04e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + 4.13e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 3.21e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 1.37e8T + 7.83e15T^{2} \) |
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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
−14.43778301766748980706854842843, −12.82666221692377846218780089022, −12.30482519381278594575372213116, −10.88404537470365571397540493417, −8.856244160389461124444777154946, −7.59772067037359018833737578576, −6.28291828572131789637808728803, −5.03757398866462639651413775776, −2.09318868283253642273734894900, −0.36267380328321007906475418488,
2.48456516006523556043329019482, 3.79120142297967987720400787063, 6.08266916843451860897245812224, 7.16743117473204548655220404605, 9.696903254933945894369661225377, 10.57324624844955030034446988086, 11.33503249157477949996689548249, 12.61115214017099124515464778997, 14.71701042016575710412289974802, 15.23209932208531841150524811318