| L(s) = 1 | + (−0.859 − 1.12i)2-s + (1.37 + 1.05i)3-s + (−0.521 + 1.93i)4-s + (−0.128 − 0.478i)5-s + (0.00940 − 2.44i)6-s + (−1.18 + 2.36i)7-s + (2.61 − 1.07i)8-s + (0.760 + 2.90i)9-s + (−0.427 + 0.555i)10-s + (−0.388 + 1.45i)11-s + (−2.75 + 2.09i)12-s + (0.486 − 0.486i)13-s + (3.67 − 0.703i)14-s + (0.330 − 0.792i)15-s + (−3.45 − 2.01i)16-s + (1.59 + 2.75i)17-s + ⋯ |
| L(s) = 1 | + (−0.607 − 0.793i)2-s + (0.791 + 0.610i)3-s + (−0.260 + 0.965i)4-s + (−0.0573 − 0.214i)5-s + (0.00383 − 0.999i)6-s + (−0.447 + 0.894i)7-s + (0.925 − 0.379i)8-s + (0.253 + 0.967i)9-s + (−0.135 + 0.175i)10-s + (−0.117 + 0.437i)11-s + (−0.796 + 0.604i)12-s + (0.135 − 0.135i)13-s + (0.982 − 0.188i)14-s + (0.0854 − 0.204i)15-s + (−0.863 − 0.503i)16-s + (0.386 + 0.669i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.757 - 0.652i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.757 - 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.05941 + 0.393408i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05941 + 0.393408i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.859 + 1.12i)T \) |
| 3 | \( 1 + (-1.37 - 1.05i)T \) |
| 7 | \( 1 + (1.18 - 2.36i)T \) |
| good | 5 | \( 1 + (0.128 + 0.478i)T + (-4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (0.388 - 1.45i)T + (-9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (-0.486 + 0.486i)T - 13iT^{2} \) |
| 17 | \( 1 + (-1.59 - 2.75i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.722 - 2.69i)T + (-16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (2.26 - 3.92i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.56 + 1.56i)T - 29iT^{2} \) |
| 31 | \( 1 + (0.851 - 0.491i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.395 + 1.47i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 - 2.87iT - 41T^{2} \) |
| 43 | \( 1 + (-5.20 + 5.20i)T - 43iT^{2} \) |
| 47 | \( 1 + (-6.34 + 10.9i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.451 + 1.68i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (-12.7 - 3.42i)T + (51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (3.12 + 11.6i)T + (-52.8 + 30.5i)T^{2} \) |
| 67 | \( 1 + (3.29 - 12.3i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 + (5.65 + 9.78i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.58 - 6.20i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (4.94 + 4.94i)T + 83iT^{2} \) |
| 89 | \( 1 + (9.31 + 5.37i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 4.48iT - 97T^{2} \) |
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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
−11.63246091294901519101944079562, −10.36426532682219584615927166889, −9.901746942678159978640240238923, −8.893526780505642396960234637795, −8.353729894436701397292217686881, −7.29630747471330941888339536011, −5.56125783415523969667315019850, −4.18667678768210668136876922098, −3.16349117701846928658972880417, −1.98958000689803376620606531140,
0.950832391623970802697028481795, 2.88549967833662180990365190437, 4.38052443988501658832527897939, 5.98648423988242905470745773125, 6.98127296810075286822243496559, 7.49897035248696420312335408329, 8.560396711651744115789845345561, 9.348416862940373704893131499068, 10.27375994099591298706314404413, 11.20990761545697885581817450515
