| L(s) = 1 | + (0.173 − 0.984i)2-s + (−1.51 − 0.833i)3-s + (−0.939 − 0.342i)4-s + (0.342 + 0.939i)5-s + (−1.08 + 1.35i)6-s + (0.534 + 0.926i)7-s + (−0.5 + 0.866i)8-s + (1.61 + 2.53i)9-s + (0.984 − 0.173i)10-s + (−1.99 − 1.15i)11-s + (1.14 + 1.30i)12-s + (2.18 + 2.60i)13-s + (1.00 − 0.365i)14-s + (0.263 − 1.71i)15-s + (0.766 + 0.642i)16-s + (4.86 + 0.857i)17-s + ⋯ |
| L(s) = 1 | + (0.122 − 0.696i)2-s + (−0.876 − 0.481i)3-s + (−0.469 − 0.171i)4-s + (0.152 + 0.420i)5-s + (−0.442 + 0.551i)6-s + (0.202 + 0.350i)7-s + (−0.176 + 0.306i)8-s + (0.537 + 0.843i)9-s + (0.311 − 0.0549i)10-s + (−0.600 − 0.346i)11-s + (0.329 + 0.375i)12-s + (0.605 + 0.721i)13-s + (0.268 − 0.0977i)14-s + (0.0680 − 0.441i)15-s + (0.191 + 0.160i)16-s + (1.17 + 0.207i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.882 + 0.469i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.882 + 0.469i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.12447 - 0.280353i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.12447 - 0.280353i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.173 + 0.984i)T \) |
| 3 | \( 1 + (1.51 + 0.833i)T \) |
| 5 | \( 1 + (-0.342 - 0.939i)T \) |
| 19 | \( 1 + (-2.31 - 3.69i)T \) |
| good | 7 | \( 1 + (-0.534 - 0.926i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.99 + 1.15i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.18 - 2.60i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-4.86 - 0.857i)T + (15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (-1.95 + 5.37i)T + (-17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.777 - 4.40i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-0.497 + 0.287i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 1.64iT - 37T^{2} \) |
| 41 | \( 1 + (4.01 + 3.37i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-8.10 + 2.94i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-3.87 + 0.683i)T + (44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (-7.98 - 2.90i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (1.85 - 10.5i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (4.66 + 1.69i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-1.24 + 0.219i)T + (62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (9.44 - 3.43i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-12.9 - 10.8i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (0.658 - 0.784i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (5.23 - 3.02i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.30 + 1.09i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (6.47 + 1.14i)T + (91.1 + 33.1i)T^{2} \) |
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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
−10.56505262956434516099923107268, −10.35536572334710820607046970296, −8.991711364517031726650879868200, −8.010362465048151173221073322234, −6.97040983581305805562911447321, −5.87661947746517262727901677136, −5.26344839631351514573624008466, −3.92144089940936519716155840284, −2.55596023881955185321930458949, −1.22390419019178064755596211422,
0.895717397907201096419282167175, 3.32745189751654850320586797480, 4.54480782550430328183904630480, 5.34901019819190534092188163113, 5.97842953653668817179930967589, 7.24207789532780410059486655787, 7.915987295627040157314331577809, 9.175542326242121413150906850756, 9.888068022189539675535544952241, 10.73884106166413843556465892935
