| L(s) = 1 | + (−1.07 + 0.923i)2-s + (1.49 − 0.879i)3-s + (0.294 − 1.97i)4-s + (−1.88 − 1.00i)5-s + (−0.786 + 2.31i)6-s + (2.66 + 0.529i)7-s + (1.51 + 2.39i)8-s + (1.45 − 2.62i)9-s + (2.94 − 0.660i)10-s + (−2.44 + 2.00i)11-s + (−1.30 − 3.21i)12-s + (0.359 − 0.192i)13-s + (−3.33 + 1.88i)14-s + (−3.69 + 0.153i)15-s + (−3.82 − 1.16i)16-s + (7.42 − 3.07i)17-s + ⋯ |
| L(s) = 1 | + (−0.757 + 0.652i)2-s + (0.861 − 0.507i)3-s + (0.147 − 0.989i)4-s + (−0.841 − 0.449i)5-s + (−0.320 + 0.947i)6-s + (1.00 + 0.200i)7-s + (0.534 + 0.845i)8-s + (0.484 − 0.874i)9-s + (0.931 − 0.208i)10-s + (−0.736 + 0.604i)11-s + (−0.375 − 0.926i)12-s + (0.0997 − 0.0533i)13-s + (−0.892 + 0.505i)14-s + (−0.953 + 0.0397i)15-s + (−0.956 − 0.291i)16-s + (1.80 − 0.746i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.865 + 0.501i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.865 + 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.15114 - 0.309210i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.15114 - 0.309210i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.07 - 0.923i)T \) |
| 3 | \( 1 + (-1.49 + 0.879i)T \) |
| good | 5 | \( 1 + (1.88 + 1.00i)T + (2.77 + 4.15i)T^{2} \) |
| 7 | \( 1 + (-2.66 - 0.529i)T + (6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (2.44 - 2.00i)T + (2.14 - 10.7i)T^{2} \) |
| 13 | \( 1 + (-0.359 + 0.192i)T + (7.22 - 10.8i)T^{2} \) |
| 17 | \( 1 + (-7.42 + 3.07i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (-2.07 + 6.84i)T + (-15.7 - 10.5i)T^{2} \) |
| 23 | \( 1 + (-0.519 - 0.347i)T + (8.80 + 21.2i)T^{2} \) |
| 29 | \( 1 + (5.63 + 4.62i)T + (5.65 + 28.4i)T^{2} \) |
| 31 | \( 1 + (-0.985 + 0.985i)T - 31iT^{2} \) |
| 37 | \( 1 + (-2.05 - 6.78i)T + (-30.7 + 20.5i)T^{2} \) |
| 41 | \( 1 + (0.894 + 0.597i)T + (15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (0.827 - 8.39i)T + (-42.1 - 8.38i)T^{2} \) |
| 47 | \( 1 + (-6.66 + 2.76i)T + (33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (9.17 - 7.53i)T + (10.3 - 51.9i)T^{2} \) |
| 59 | \( 1 + (-9.35 - 5.00i)T + (32.7 + 49.0i)T^{2} \) |
| 61 | \( 1 + (-0.741 - 7.52i)T + (-59.8 + 11.9i)T^{2} \) |
| 67 | \( 1 + (-3.75 + 0.370i)T + (65.7 - 13.0i)T^{2} \) |
| 71 | \( 1 + (1.09 + 0.217i)T + (65.5 + 27.1i)T^{2} \) |
| 73 | \( 1 + (-0.726 - 3.65i)T + (-67.4 + 27.9i)T^{2} \) |
| 79 | \( 1 + (4.98 - 12.0i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (-0.0763 + 0.251i)T + (-69.0 - 46.1i)T^{2} \) |
| 89 | \( 1 + (0.606 + 0.907i)T + (-34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (-3.22 - 3.22i)T + 97iT^{2} \) |
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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
−11.35697568861968138733687922240, −9.969895578284201890901475453641, −9.226508308562145320199088034771, −8.120252458907582087488838410559, −7.82073556200363800340745704775, −7.06659853072740106089249427197, −5.46992845068945766548681417967, −4.49414454314453683743978505337, −2.65055542557470327727642322687, −1.06026998582529078454107925289,
1.70240647996088969725983864635, 3.34033202837119720179444529263, 3.80956458524111852193196396987, 5.37989083342557512047520111222, 7.50855324776599481042578573210, 7.85074653191904625356867111721, 8.497664218324401421631041140982, 9.718320763456829119501494252818, 10.55675472875367902230343853277, 11.07467530984915624030763416328
