| L(s) = 1 | + (1.19 + 0.759i)2-s + (1.48 − 0.899i)3-s + (0.845 + 1.81i)4-s + (1.53 − 2.86i)5-s + (2.44 + 0.0512i)6-s + (−4.19 − 0.834i)7-s + (−0.368 + 2.80i)8-s + (1.38 − 2.66i)9-s + (4.00 − 2.25i)10-s + (1.80 + 2.20i)11-s + (2.88 + 1.92i)12-s + (1.76 + 3.29i)13-s + (−4.37 − 4.18i)14-s + (−0.311 − 5.61i)15-s + (−2.56 + 3.06i)16-s + (3.65 − 1.51i)17-s + ⋯ |
| L(s) = 1 | + (0.843 + 0.537i)2-s + (0.854 − 0.519i)3-s + (0.422 + 0.906i)4-s + (0.685 − 1.28i)5-s + (0.999 + 0.0209i)6-s + (−1.58 − 0.315i)7-s + (−0.130 + 0.991i)8-s + (0.460 − 0.887i)9-s + (1.26 − 0.712i)10-s + (0.545 + 0.664i)11-s + (0.832 + 0.554i)12-s + (0.488 + 0.913i)13-s + (−1.16 − 1.11i)14-s + (−0.0803 − 1.45i)15-s + (−0.642 + 0.766i)16-s + (0.886 − 0.367i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0366i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0366i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.74492 - 0.0503853i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.74492 - 0.0503853i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.19 - 0.759i)T \) |
| 3 | \( 1 + (-1.48 + 0.899i)T \) |
| good | 5 | \( 1 + (-1.53 + 2.86i)T + (-2.77 - 4.15i)T^{2} \) |
| 7 | \( 1 + (4.19 + 0.834i)T + (6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (-1.80 - 2.20i)T + (-2.14 + 10.7i)T^{2} \) |
| 13 | \( 1 + (-1.76 - 3.29i)T + (-7.22 + 10.8i)T^{2} \) |
| 17 | \( 1 + (-3.65 + 1.51i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (3.04 + 0.924i)T + (15.7 + 10.5i)T^{2} \) |
| 23 | \( 1 + (0.756 + 0.505i)T + (8.80 + 21.2i)T^{2} \) |
| 29 | \( 1 + (1.91 - 2.33i)T + (-5.65 - 28.4i)T^{2} \) |
| 31 | \( 1 + (6.48 - 6.48i)T - 31iT^{2} \) |
| 37 | \( 1 + (8.40 - 2.54i)T + (30.7 - 20.5i)T^{2} \) |
| 41 | \( 1 + (3.63 + 2.43i)T + (15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (-6.98 - 0.687i)T + (42.1 + 8.38i)T^{2} \) |
| 47 | \( 1 + (-6.43 + 2.66i)T + (33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (3.44 + 4.19i)T + (-10.3 + 51.9i)T^{2} \) |
| 59 | \( 1 + (4.34 - 8.13i)T + (-32.7 - 49.0i)T^{2} \) |
| 61 | \( 1 + (-10.8 + 1.06i)T + (59.8 - 11.9i)T^{2} \) |
| 67 | \( 1 + (-0.0832 - 0.845i)T + (-65.7 + 13.0i)T^{2} \) |
| 71 | \( 1 + (14.9 + 2.97i)T + (65.5 + 27.1i)T^{2} \) |
| 73 | \( 1 + (-2.01 - 10.1i)T + (-67.4 + 27.9i)T^{2} \) |
| 79 | \( 1 + (-2.46 + 5.94i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (3.77 + 1.14i)T + (69.0 + 46.1i)T^{2} \) |
| 89 | \( 1 + (1.95 + 2.92i)T + (-34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (-1.40 - 1.40i)T + 97iT^{2} \) |
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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
−11.98942109251672878889232501111, −10.14093910435427542407937124138, −9.061336213427786314679856149007, −8.803381917085050275011391872265, −7.23701394725424800201129252192, −6.67289974435381269358983059970, −5.62893498382583350627088555886, −4.26475243505167587589193294138, −3.33290536962724395819174221500, −1.76875042810153082968017341146,
2.28091021347347935604653334725, 3.31150596001078873160901616562, 3.69104206700874920738497828052, 5.75583991868752893689471912336, 6.19456244352711454420197861230, 7.37977569237318934113137531572, 8.977005848474516146770805075672, 9.839550167534268110694672466440, 10.37742271647121960066344244922, 11.08721065689441372086477733708
