L(s) = 1 | + (1.64 + 0.550i)3-s + (−1.00 − 2.76i)5-s + (0.201 − 0.0355i)7-s + (2.39 + 1.80i)9-s + (2.91 + 1.05i)11-s + (1.23 + 1.03i)13-s + (−0.130 − 5.09i)15-s + (−2.86 − 1.65i)17-s + (5.98 − 3.45i)19-s + (0.350 + 0.0525i)21-s + (1.21 − 6.89i)23-s + (−2.79 + 2.34i)25-s + (2.93 + 4.28i)27-s + (−2.70 − 3.22i)29-s + (−0.173 − 0.0305i)31-s + ⋯ |
L(s) = 1 | + (0.948 + 0.317i)3-s + (−0.449 − 1.23i)5-s + (0.0760 − 0.0134i)7-s + (0.797 + 0.602i)9-s + (0.877 + 0.319i)11-s + (0.342 + 0.287i)13-s + (−0.0336 − 1.31i)15-s + (−0.694 − 0.401i)17-s + (1.37 − 0.792i)19-s + (0.0764 + 0.0114i)21-s + (0.253 − 1.43i)23-s + (−0.559 + 0.469i)25-s + (0.564 + 0.825i)27-s + (−0.503 − 0.599i)29-s + (−0.0311 − 0.00548i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.378i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.925 + 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.83877 - 0.361946i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.83877 - 0.361946i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.64 - 0.550i)T \) |
good | 5 | \( 1 + (1.00 + 2.76i)T + (-3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.201 + 0.0355i)T + (6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (-2.91 - 1.05i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-1.23 - 1.03i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (2.86 + 1.65i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-5.98 + 3.45i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.21 + 6.89i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (2.70 + 3.22i)T + (-5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (0.173 + 0.0305i)T + (29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (5.46 - 9.46i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.982 - 1.17i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (3.57 - 9.83i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.76 - 9.98i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + 3.56iT - 53T^{2} \) |
| 59 | \( 1 + (4.69 - 1.71i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.18 - 6.69i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (2.60 - 3.10i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-2.61 + 4.52i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (4.68 + 8.11i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (6.13 + 7.31i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (8.47 - 7.10i)T + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-2.64 + 1.52i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.58 + 0.577i)T + (74.3 + 62.3i)T^{2} \) |
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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
−11.21315375665235255169585589789, −9.857953267700444997560678574234, −9.123862253187246346264442789924, −8.585676176916799812940239188516, −7.64447349105714577775366819500, −6.56200420865343843756964615087, −4.80703583757921108857200851901, −4.40470713699698319555594516031, −3.02765889873010073125619366014, −1.35635775811634950409251378580,
1.76014425560101709044188695889, 3.37146274638413071891098778087, 3.69860478246026821800670151872, 5.65145227100512272930978429995, 6.93274385766506706465617985090, 7.35792557277206164492445004627, 8.452194158735912212109503610158, 9.309465189569863463320053237328, 10.29511712554824505010542126642, 11.24249059332575219962976368912