| L(s) = 1 | + (0.368 + 0.236i)3-s + (5.43 + 2.48i)5-s + (−1.13 + 3.86i)7-s + (−3.65 − 8.01i)9-s + (13.7 + 11.9i)11-s + (17.4 − 5.11i)13-s + (1.41 + 2.19i)15-s + (−15.1 + 2.18i)17-s + (−20.8 − 2.99i)19-s + (−1.33 + 1.15i)21-s + (−18.5 − 13.6i)23-s + (6.99 + 8.07i)25-s + (1.10 − 7.71i)27-s + (−5.30 − 36.8i)29-s + (−16.4 + 10.5i)31-s + ⋯ |
| L(s) = 1 | + (0.122 + 0.0788i)3-s + (1.08 + 0.496i)5-s + (−0.162 + 0.552i)7-s + (−0.406 − 0.890i)9-s + (1.25 + 1.08i)11-s + (1.34 − 0.393i)13-s + (0.0942 + 0.146i)15-s + (−0.893 + 0.128i)17-s + (−1.09 − 0.157i)19-s + (−0.0635 + 0.0550i)21-s + (−0.806 − 0.591i)23-s + (0.279 + 0.322i)25-s + (0.0410 − 0.285i)27-s + (−0.182 − 1.27i)29-s + (−0.530 + 0.340i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.924 - 0.381i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.924 - 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.52767 + 0.302941i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.52767 + 0.302941i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 + (18.5 + 13.6i)T \) |
| good | 3 | \( 1 + (-0.368 - 0.236i)T + (3.73 + 8.18i)T^{2} \) |
| 5 | \( 1 + (-5.43 - 2.48i)T + (16.3 + 18.8i)T^{2} \) |
| 7 | \( 1 + (1.13 - 3.86i)T + (-41.2 - 26.4i)T^{2} \) |
| 11 | \( 1 + (-13.7 - 11.9i)T + (17.2 + 119. i)T^{2} \) |
| 13 | \( 1 + (-17.4 + 5.11i)T + (142. - 91.3i)T^{2} \) |
| 17 | \( 1 + (15.1 - 2.18i)T + (277. - 81.4i)T^{2} \) |
| 19 | \( 1 + (20.8 + 2.99i)T + (346. + 101. i)T^{2} \) |
| 29 | \( 1 + (5.30 + 36.8i)T + (-806. + 236. i)T^{2} \) |
| 31 | \( 1 + (16.4 - 10.5i)T + (399. - 874. i)T^{2} \) |
| 37 | \( 1 + (27.3 - 12.4i)T + (896. - 1.03e3i)T^{2} \) |
| 41 | \( 1 + (-23.9 + 52.5i)T + (-1.10e3 - 1.27e3i)T^{2} \) |
| 43 | \( 1 + (25.3 - 39.5i)T + (-768. - 1.68e3i)T^{2} \) |
| 47 | \( 1 - 3.94T + 2.20e3T^{2} \) |
| 53 | \( 1 + (-1.98 + 6.74i)T + (-2.36e3 - 1.51e3i)T^{2} \) |
| 59 | \( 1 + (-1.71 + 0.504i)T + (2.92e3 - 1.88e3i)T^{2} \) |
| 61 | \( 1 + (-21.0 - 32.8i)T + (-1.54e3 + 3.38e3i)T^{2} \) |
| 67 | \( 1 + (-85.9 + 74.4i)T + (638. - 4.44e3i)T^{2} \) |
| 71 | \( 1 + (53.2 + 61.5i)T + (-717. + 4.98e3i)T^{2} \) |
| 73 | \( 1 + (10.2 - 71.2i)T + (-5.11e3 - 1.50e3i)T^{2} \) |
| 79 | \( 1 + (-0.614 - 2.09i)T + (-5.25e3 + 3.37e3i)T^{2} \) |
| 83 | \( 1 + (5.17 - 2.36i)T + (4.51e3 - 5.20e3i)T^{2} \) |
| 89 | \( 1 + (-45.2 + 70.4i)T + (-3.29e3 - 7.20e3i)T^{2} \) |
| 97 | \( 1 + (-134. - 61.3i)T + (6.16e3 + 7.11e3i)T^{2} \) |
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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
−13.99679062033747103541821020268, −12.86532859061766414789578111582, −11.80564481467336773854809222720, −10.53093619776422626448165496145, −9.421381913278711383859007395237, −8.654104066176325810687443784873, −6.56035589863482924473499554407, −6.06848776332429664371548564484, −3.97814717687103361477474921969, −2.13528258930086671903952288355,
1.68581435424895698147896811330, 3.88560992869744768315727616594, 5.64625867223780765170739781891, 6.62918216023565637306731088153, 8.478589259457221548063723519496, 9.142379062796572827266093189102, 10.60566315681873701216282211681, 11.42989804171574228906300876520, 13.12374140250827707891934109690, 13.66295608773667716168600936745
