| L(s) = 1 | + (−1.39 − 0.252i)2-s + (−0.540 + 0.841i)3-s + (1.87 + 0.702i)4-s + (−2.13 − 0.975i)5-s + (0.964 − 1.03i)6-s + (0.595 + 0.174i)7-s + (−2.42 − 1.45i)8-s + (−0.415 − 0.909i)9-s + (2.72 + 1.89i)10-s + (1.11 + 0.963i)11-s + (−1.60 + 1.19i)12-s + (0.995 + 3.38i)13-s + (−0.784 − 0.393i)14-s + (1.97 − 1.26i)15-s + (3.01 + 2.63i)16-s + (−1.11 − 7.76i)17-s + ⋯ |
| L(s) = 1 | + (−0.983 − 0.178i)2-s + (−0.312 + 0.485i)3-s + (0.936 + 0.351i)4-s + (−0.954 − 0.436i)5-s + (0.393 − 0.422i)6-s + (0.225 + 0.0661i)7-s + (−0.858 − 0.513i)8-s + (−0.138 − 0.303i)9-s + (0.861 + 0.599i)10-s + (0.335 + 0.290i)11-s + (−0.462 + 0.345i)12-s + (0.276 + 0.940i)13-s + (−0.209 − 0.105i)14-s + (0.509 − 0.327i)15-s + (0.752 + 0.658i)16-s + (−0.270 − 1.88i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.464 + 0.885i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.464 + 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.495703 - 0.299746i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.495703 - 0.299746i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.39 + 0.252i)T \) |
| 3 | \( 1 + (0.540 - 0.841i)T \) |
| 23 | \( 1 + (-4.79 + 0.0498i)T \) |
| good | 5 | \( 1 + (2.13 + 0.975i)T + (3.27 + 3.77i)T^{2} \) |
| 7 | \( 1 + (-0.595 - 0.174i)T + (5.88 + 3.78i)T^{2} \) |
| 11 | \( 1 + (-1.11 - 0.963i)T + (1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.995 - 3.38i)T + (-10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (1.11 + 7.76i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (6.09 + 0.876i)T + (18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (1.45 - 0.209i)T + (27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (-5.62 + 3.61i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (-9.90 + 4.52i)T + (24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (1.33 - 2.92i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-5.83 + 9.08i)T + (-17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 + (0.599 - 2.04i)T + (-44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (2.33 + 7.95i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (6.43 + 10.0i)T + (-25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (-8.28 + 7.18i)T + (9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (5.25 + 6.06i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (1.40 - 9.73i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (-4.20 + 1.23i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (1.42 - 0.652i)T + (54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (-5.06 - 3.25i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (4.42 - 9.67i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−10.79598285194225809020348509980, −9.463061175612339582062870659703, −9.101698658395803758656129498551, −8.117417994176471672171231670780, −7.18856548968628376042391898597, −6.33562203896804864470618260341, −4.77588404175027265711179427171, −3.95769065239994541369467691706, −2.42378296669046355395674393756, −0.55018303679081700588019457767,
1.21012764981506907524709108231, 2.82519086125317928376685631072, 4.19194818780705206563991719627, 5.89925472697996649040471257947, 6.49257231771428247799677563120, 7.59080574799972954265364738326, 8.173149154368158682638021512413, 8.869515702622577982797750480530, 10.36545477307208581342854683342, 10.83220972002987003906313526957
