| L(s) = 1 | + (0.151 + 2.01i)2-s + (−2.06 + 0.310i)4-s + (1.66 − 1.54i)5-s + (−1.77 + 1.95i)7-s + (−0.0388 − 0.170i)8-s + (3.37 + 3.12i)10-s + (−1.57 + 1.07i)11-s + (4.29 + 2.06i)13-s + (−4.21 − 3.28i)14-s + (−3.65 + 1.12i)16-s + (1.47 + 3.76i)17-s + (0.218 + 0.379i)19-s + (−2.95 + 3.71i)20-s + (−2.40 − 3.01i)22-s + (−2.49 + 6.35i)23-s + ⋯ |
| L(s) = 1 | + (0.106 + 1.42i)2-s + (−1.03 + 0.155i)4-s + (0.745 − 0.692i)5-s + (−0.671 + 0.740i)7-s + (−0.0137 − 0.0601i)8-s + (1.06 + 0.989i)10-s + (−0.475 + 0.324i)11-s + (1.19 + 0.573i)13-s + (−1.12 − 0.878i)14-s + (−0.912 + 0.281i)16-s + (0.358 + 0.914i)17-s + (0.0502 + 0.0869i)19-s + (−0.661 + 0.829i)20-s + (−0.512 − 0.642i)22-s + (−0.519 + 1.32i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.789 - 0.614i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.789 - 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.489298 + 1.42504i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.489298 + 1.42504i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + (1.77 - 1.95i)T \) |
| good | 2 | \( 1 + (-0.151 - 2.01i)T + (-1.97 + 0.298i)T^{2} \) |
| 5 | \( 1 + (-1.66 + 1.54i)T + (0.373 - 4.98i)T^{2} \) |
| 11 | \( 1 + (1.57 - 1.07i)T + (4.01 - 10.2i)T^{2} \) |
| 13 | \( 1 + (-4.29 - 2.06i)T + (8.10 + 10.1i)T^{2} \) |
| 17 | \( 1 + (-1.47 - 3.76i)T + (-12.4 + 11.5i)T^{2} \) |
| 19 | \( 1 + (-0.218 - 0.379i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.49 - 6.35i)T + (-16.8 - 15.6i)T^{2} \) |
| 29 | \( 1 + (-5.30 + 6.64i)T + (-6.45 - 28.2i)T^{2} \) |
| 31 | \( 1 + (0.409 - 0.709i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.68 - 0.555i)T + (35.3 + 10.9i)T^{2} \) |
| 41 | \( 1 + (2.40 + 10.5i)T + (-36.9 + 17.7i)T^{2} \) |
| 43 | \( 1 + (-1.79 + 7.87i)T + (-38.7 - 18.6i)T^{2} \) |
| 47 | \( 1 + (0.114 + 1.53i)T + (-46.4 + 7.00i)T^{2} \) |
| 53 | \( 1 + (0.818 - 0.123i)T + (50.6 - 15.6i)T^{2} \) |
| 59 | \( 1 + (-2.23 - 2.07i)T + (4.40 + 58.8i)T^{2} \) |
| 61 | \( 1 + (-0.0576 - 0.00869i)T + (58.2 + 17.9i)T^{2} \) |
| 67 | \( 1 + (-6.06 + 10.5i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3.05 + 3.83i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (0.809 - 10.8i)T + (-72.1 - 10.8i)T^{2} \) |
| 79 | \( 1 + (1.22 + 2.13i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.16 + 2.00i)T + (51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (-3.35 - 2.29i)T + (32.5 + 82.8i)T^{2} \) |
| 97 | \( 1 - 4.05T + 97T^{2} \) |
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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
−11.62132115501161205858018322126, −10.29031999006353169275107963953, −9.303945437719337364634574254224, −8.662335564850960829392131825164, −7.80442427197667290217519373873, −6.58249148695589062449577191341, −5.85087877528779769501049153157, −5.31327246967113361290577338754, −3.89077754813450620474795593767, −1.98589368050962046590218665416,
0.978861677860748027815994495847, 2.68881618443110260864800927826, 3.27729415835504550202377065315, 4.57255127008308830200533576021, 6.08736216355566331631732410739, 6.85748239435742717393115732150, 8.232531223844702830629761441884, 9.482909187354418576859615767711, 10.21233746727532469933507941532, 10.67045290066968874570734768144
