L(s) = 1 | + (−0.700 + 1.78i)2-s + (−1.50 + 0.851i)3-s + (−1.22 − 1.13i)4-s + (2.53 − 1.22i)5-s + (−0.462 − 3.28i)6-s + (−2.05 − 1.66i)7-s + (−0.560 + 0.269i)8-s + (1.55 − 2.56i)9-s + (0.403 + 5.38i)10-s + (0.0782 + 0.0980i)11-s + (2.82 + 0.673i)12-s + (1.84 − 4.69i)13-s + (4.40 − 2.51i)14-s + (−2.78 + 4.00i)15-s + (−0.339 − 4.53i)16-s + (3.92 − 3.64i)17-s + ⋯ |
L(s) = 1 | + (−0.495 + 1.26i)2-s + (−0.870 + 0.491i)3-s + (−0.614 − 0.569i)4-s + (1.13 − 0.546i)5-s + (−0.189 − 1.34i)6-s + (−0.778 − 0.627i)7-s + (−0.198 + 0.0954i)8-s + (0.516 − 0.856i)9-s + (0.127 + 1.70i)10-s + (0.0235 + 0.0295i)11-s + (0.814 + 0.194i)12-s + (0.510 − 1.30i)13-s + (1.17 − 0.671i)14-s + (−0.719 + 1.03i)15-s + (−0.0848 − 1.13i)16-s + (0.952 − 0.883i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.901 - 0.432i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.901 - 0.432i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.792554 + 0.180298i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.792554 + 0.180298i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.50 - 0.851i)T \) |
| 7 | \( 1 + (2.05 + 1.66i)T \) |
good | 2 | \( 1 + (0.700 - 1.78i)T + (-1.46 - 1.36i)T^{2} \) |
| 5 | \( 1 + (-2.53 + 1.22i)T + (3.11 - 3.90i)T^{2} \) |
| 11 | \( 1 + (-0.0782 - 0.0980i)T + (-2.44 + 10.7i)T^{2} \) |
| 13 | \( 1 + (-1.84 + 4.69i)T + (-9.52 - 8.84i)T^{2} \) |
| 17 | \( 1 + (-3.92 + 3.64i)T + (1.27 - 16.9i)T^{2} \) |
| 19 | \( 1 + (0.421 - 0.729i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.52 + 6.68i)T + (-20.7 - 9.97i)T^{2} \) |
| 29 | \( 1 + (-0.484 - 0.149i)T + (23.9 + 16.3i)T^{2} \) |
| 31 | \( 1 + (5.01 - 8.68i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4.98 + 1.53i)T + (30.5 + 20.8i)T^{2} \) |
| 41 | \( 1 + (-3.97 - 2.71i)T + (14.9 + 38.1i)T^{2} \) |
| 43 | \( 1 + (1.27 - 0.868i)T + (15.7 - 40.0i)T^{2} \) |
| 47 | \( 1 + (-3.10 + 7.89i)T + (-34.4 - 31.9i)T^{2} \) |
| 53 | \( 1 + (-0.276 + 0.0853i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (-4.55 + 3.10i)T + (21.5 - 54.9i)T^{2} \) |
| 61 | \( 1 + (-2.24 + 2.08i)T + (4.55 - 60.8i)T^{2} \) |
| 67 | \( 1 + (-6.95 + 12.0i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.117 - 0.514i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (7.59 + 1.14i)T + (69.7 + 21.5i)T^{2} \) |
| 79 | \( 1 + (-0.119 - 0.207i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.530 - 1.35i)T + (-60.8 + 56.4i)T^{2} \) |
| 89 | \( 1 + (0.315 + 0.802i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (8.79 - 15.2i)T + (-48.5 - 84.0i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.76249079538466154923297017345, −10.08453146918972765167090857830, −9.391549343812806990614721682733, −8.513787646823269920413898593976, −7.23191538397398746996082952548, −6.45838298101129606849755313215, −5.62428834674724509412488704707, −5.05207590739826489433629110740, −3.29098743397248273615909149866, −0.71050304502654513886679871305,
1.50847981079479859429332901891, 2.38308482303164363724402927097, 3.77610357251576257699536619850, 5.74667116107212846484275085468, 6.11720819196463101671667020551, 7.18284030343704235278516976810, 8.803739068099451547290573973771, 9.670975440110463911356907978285, 10.15796766882918173310762376788, 11.15360229942199068010194625147