| L(s) = 1 | + (2.02 + 1.78i)2-s + (−0.305 + 1.70i)3-s + (0.668 + 5.25i)4-s + (−0.249 − 0.0362i)5-s + (−3.66 + 2.91i)6-s + (−0.0230 − 0.254i)7-s + (−5.00 + 7.38i)8-s + (−2.81 − 1.04i)9-s + (−0.440 − 0.519i)10-s + (2.39 − 2.43i)11-s + (−9.17 − 0.469i)12-s + (4.36 + 0.157i)13-s + (0.407 − 0.556i)14-s + (0.138 − 0.414i)15-s + (−13.0 + 3.37i)16-s + (3.26 − 1.51i)17-s + ⋯ |
| L(s) = 1 | + (1.43 + 1.26i)2-s + (−0.176 + 0.984i)3-s + (0.334 + 2.62i)4-s + (−0.111 − 0.0162i)5-s + (−1.49 + 1.18i)6-s + (−0.00870 − 0.0961i)7-s + (−1.77 + 2.61i)8-s + (−0.937 − 0.347i)9-s + (−0.139 − 0.164i)10-s + (0.721 − 0.734i)11-s + (−2.64 − 0.135i)12-s + (1.21 + 0.0437i)13-s + (0.108 − 0.148i)14-s + (0.0356 − 0.106i)15-s + (−3.26 + 0.844i)16-s + (0.792 − 0.366i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0647i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0647i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0912155 + 2.81429i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0912155 + 2.81429i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.305 - 1.70i)T \) |
| 59 | \( 1 + (3.36 + 6.90i)T \) |
| good | 2 | \( 1 + (-2.02 - 1.78i)T + (0.252 + 1.98i)T^{2} \) |
| 5 | \( 1 + (0.249 + 0.0362i)T + (4.79 + 1.42i)T^{2} \) |
| 7 | \( 1 + (0.0230 + 0.254i)T + (-6.88 + 1.25i)T^{2} \) |
| 11 | \( 1 + (-2.39 + 2.43i)T + (-0.198 - 10.9i)T^{2} \) |
| 13 | \( 1 + (-4.36 - 0.157i)T + (12.9 + 0.938i)T^{2} \) |
| 17 | \( 1 + (-3.26 + 1.51i)T + (11.0 - 12.9i)T^{2} \) |
| 19 | \( 1 + (3.45 + 0.760i)T + (17.2 + 7.97i)T^{2} \) |
| 23 | \( 1 + (-0.916 - 0.346i)T + (17.2 + 15.2i)T^{2} \) |
| 29 | \( 1 + (-0.634 - 3.15i)T + (-26.7 + 11.2i)T^{2} \) |
| 31 | \( 1 + (1.15 - 3.62i)T + (-25.3 - 17.8i)T^{2} \) |
| 37 | \( 1 + (2.27 + 3.35i)T + (-13.6 + 34.3i)T^{2} \) |
| 41 | \( 1 + (-5.09 - 4.17i)T + (8.08 + 40.1i)T^{2} \) |
| 43 | \( 1 + (-9.77 + 2.52i)T + (37.6 - 20.8i)T^{2} \) |
| 47 | \( 1 + (5.55 - 0.807i)T + (45.0 - 13.3i)T^{2} \) |
| 53 | \( 1 + (4.71 - 5.54i)T + (-8.57 - 52.3i)T^{2} \) |
| 61 | \( 1 + (1.36 + 1.20i)T + (7.68 + 60.5i)T^{2} \) |
| 67 | \( 1 + (2.84 + 5.86i)T + (-41.5 + 52.5i)T^{2} \) |
| 71 | \( 1 + (4.54 - 11.3i)T + (-51.5 - 48.8i)T^{2} \) |
| 73 | \( 1 + (13.8 + 1.51i)T + (71.2 + 15.6i)T^{2} \) |
| 79 | \( 1 + (-0.314 + 17.4i)T + (-78.9 - 2.85i)T^{2} \) |
| 83 | \( 1 + (-0.210 + 0.106i)T + (49.0 - 66.9i)T^{2} \) |
| 89 | \( 1 + (-2.01 - 0.677i)T + (70.8 + 53.8i)T^{2} \) |
| 97 | \( 1 + (-2.91 - 3.98i)T + (-29.3 + 92.4i)T^{2} \) |
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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
−11.46180087138249674335949447144, −10.68686088694608483089461868657, −9.112708921234361922465362114819, −8.510311022819558005932864526383, −7.45393600248057442086577842050, −6.19929649943422264747984789881, −5.85122611165086953860702945036, −4.69042365310240738152976113409, −3.85659037102599552979158014485, −3.16568060512494951773177101461,
1.23441325593929005724172717438, 2.21527110432161051554524171405, 3.52458247668145208673342276658, 4.42470553536946208957282666447, 5.83950461966573315283070678061, 6.17099396046679850279216740335, 7.43132043008905299298361047189, 8.800318721494004203255381744773, 9.932800353186506566079599063543, 10.88096086063475999634203206127
