| L(s) = 1 | + (0.337 − 0.774i)2-s + (−1.60 − 0.646i)3-s + (0.874 + 0.942i)4-s + (−3.34 − 0.504i)5-s + (−1.04 + 1.02i)6-s + (−1.95 − 1.81i)7-s + (2.62 − 0.917i)8-s + (2.16 + 2.07i)9-s + (−1.52 + 2.42i)10-s + (−4.07 + 3.51i)11-s + (−0.795 − 2.07i)12-s + (−2.27 + 3.33i)13-s + (−2.06 + 0.900i)14-s + (5.05 + 2.97i)15-s + (−0.0167 + 0.224i)16-s + (−4.02 − 4.02i)17-s + ⋯ |
| L(s) = 1 | + (0.238 − 0.547i)2-s + (−0.927 − 0.373i)3-s + (0.437 + 0.471i)4-s + (−1.49 − 0.225i)5-s + (−0.426 + 0.418i)6-s + (−0.738 − 0.685i)7-s + (0.926 − 0.324i)8-s + (0.721 + 0.692i)9-s + (−0.481 + 0.766i)10-s + (−1.22 + 1.05i)11-s + (−0.229 − 0.600i)12-s + (−0.631 + 0.926i)13-s + (−0.551 + 0.240i)14-s + (1.30 + 0.768i)15-s + (−0.00419 + 0.0560i)16-s + (−0.975 − 0.975i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.764 - 0.644i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.764 - 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00653483 + 0.0179028i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00653483 + 0.0179028i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.60 + 0.646i)T \) |
| 29 | \( 1 + (-5.34 + 0.678i)T \) |
| good | 2 | \( 1 + (-0.337 + 0.774i)T + (-1.36 - 1.46i)T^{2} \) |
| 5 | \( 1 + (3.34 + 0.504i)T + (4.77 + 1.47i)T^{2} \) |
| 7 | \( 1 + (1.95 + 1.81i)T + (0.523 + 6.98i)T^{2} \) |
| 11 | \( 1 + (4.07 - 3.51i)T + (1.63 - 10.8i)T^{2} \) |
| 13 | \( 1 + (2.27 - 3.33i)T + (-4.74 - 12.1i)T^{2} \) |
| 17 | \( 1 + (4.02 + 4.02i)T + 17iT^{2} \) |
| 19 | \( 1 + (-0.0254 - 0.0160i)T + (8.24 + 17.1i)T^{2} \) |
| 23 | \( 1 + (6.17 + 2.42i)T + (16.8 + 15.6i)T^{2} \) |
| 31 | \( 1 + (-2.32 + 1.71i)T + (9.13 - 29.6i)T^{2} \) |
| 37 | \( 1 + (0.331 + 0.947i)T + (-28.9 + 23.0i)T^{2} \) |
| 41 | \( 1 + (1.80 + 0.483i)T + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (-0.869 + 1.17i)T + (-12.6 - 41.0i)T^{2} \) |
| 47 | \( 1 + (4.17 + 4.85i)T + (-7.00 + 46.4i)T^{2} \) |
| 53 | \( 1 + (-3.94 - 3.14i)T + (11.7 + 51.6i)T^{2} \) |
| 59 | \( 1 + (-3.11 + 1.79i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (0.0471 - 0.00176i)T + (60.8 - 4.55i)T^{2} \) |
| 67 | \( 1 + (15.6 - 1.16i)T + (66.2 - 9.98i)T^{2} \) |
| 71 | \( 1 + (4.81 - 2.31i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (0.355 - 3.15i)T + (-71.1 - 16.2i)T^{2} \) |
| 79 | \( 1 + (-0.115 + 0.612i)T + (-73.5 - 28.8i)T^{2} \) |
| 83 | \( 1 + (2.29 - 7.43i)T + (-68.5 - 46.7i)T^{2} \) |
| 89 | \( 1 + (9.35 - 1.05i)T + (86.7 - 19.8i)T^{2} \) |
| 97 | \( 1 + (-7.40 - 14.0i)T + (-54.6 + 80.1i)T^{2} \) |
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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
−11.82355474402545644155458449018, −10.69346457327281111449919002249, −9.968287486503793300449496200874, −8.084032609976527368129811198380, −7.24605318287041903495972560381, −6.79629758407893321051323770422, −4.69818076399505929526473682463, −4.14690486467655864276573139934, −2.43271077022427195488979089014, −0.01420113177883013513674242809,
3.05943569708808466001378041356, 4.51980832535081444661776513519, 5.65363438036734083493899439096, 6.35746487762279458323492066661, 7.53730846405745289121007691134, 8.396494296730863864614655161832, 10.13756532947752850095557759217, 10.70589734875010477197422178520, 11.56428972876286049615235718928, 12.35291929524414668417881657439
