| L(s) = 1 | + (0.953 − 0.590i)2-s + (−0.273 − 0.961i)3-s + (−0.331 + 0.665i)4-s + (−3.48 + 1.35i)5-s + (−0.828 − 0.755i)6-s + (−0.264 + 2.85i)7-s + (0.283 + 3.06i)8-s + (−0.850 + 0.526i)9-s + (−2.52 + 3.34i)10-s + (−2.73 + 1.69i)11-s + (0.730 + 0.136i)12-s + (0.635 − 6.85i)13-s + (1.43 + 2.87i)14-s + (2.25 + 2.98i)15-s + (1.18 + 1.56i)16-s + (−3.36 + 3.06i)17-s + ⋯ |
| L(s) = 1 | + (0.673 − 0.417i)2-s + (−0.157 − 0.555i)3-s + (−0.165 + 0.332i)4-s + (−1.55 + 0.603i)5-s + (−0.338 − 0.308i)6-s + (−0.100 + 1.07i)7-s + (0.100 + 1.08i)8-s + (−0.283 + 0.175i)9-s + (−0.798 + 1.05i)10-s + (−0.823 + 0.510i)11-s + (0.210 + 0.0394i)12-s + (0.176 − 1.90i)13-s + (0.383 + 0.769i)14-s + (0.581 + 0.770i)15-s + (0.295 + 0.391i)16-s + (−0.816 + 0.744i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.493467 + 0.587862i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.493467 + 0.587862i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.273 + 0.961i)T \) |
| 103 | \( 1 + (-9.68 - 3.04i)T \) |
| good | 2 | \( 1 + (-0.953 + 0.590i)T + (0.891 - 1.79i)T^{2} \) |
| 5 | \( 1 + (3.48 - 1.35i)T + (3.69 - 3.36i)T^{2} \) |
| 7 | \( 1 + (0.264 - 2.85i)T + (-6.88 - 1.28i)T^{2} \) |
| 11 | \( 1 + (2.73 - 1.69i)T + (4.90 - 9.84i)T^{2} \) |
| 13 | \( 1 + (-0.635 + 6.85i)T + (-12.7 - 2.38i)T^{2} \) |
| 17 | \( 1 + (3.36 - 3.06i)T + (1.56 - 16.9i)T^{2} \) |
| 19 | \( 1 + (0.242 - 0.853i)T + (-16.1 - 10.0i)T^{2} \) |
| 23 | \( 1 + (-6.61 - 4.09i)T + (10.2 + 20.5i)T^{2} \) |
| 29 | \( 1 + (0.972 - 0.376i)T + (21.4 - 19.5i)T^{2} \) |
| 31 | \( 1 + (-0.378 - 0.501i)T + (-8.48 + 29.8i)T^{2} \) |
| 37 | \( 1 + (3.90 + 0.730i)T + (34.5 + 13.3i)T^{2} \) |
| 41 | \( 1 + (-7.57 - 2.93i)T + (30.2 + 27.6i)T^{2} \) |
| 43 | \( 1 + (0.298 - 0.0557i)T + (40.0 - 15.5i)T^{2} \) |
| 47 | \( 1 + 4.10T + 47T^{2} \) |
| 53 | \( 1 + (3.61 - 12.7i)T + (-45.0 - 27.9i)T^{2} \) |
| 59 | \( 1 + (0.797 + 8.60i)T + (-57.9 + 10.8i)T^{2} \) |
| 61 | \( 1 + (7.92 - 7.22i)T + (5.62 - 60.7i)T^{2} \) |
| 67 | \( 1 + (-0.895 - 9.66i)T + (-65.8 + 12.3i)T^{2} \) |
| 71 | \( 1 + (1.72 + 0.666i)T + (52.4 + 47.8i)T^{2} \) |
| 73 | \( 1 + (-4.50 - 1.74i)T + (53.9 + 49.1i)T^{2} \) |
| 79 | \( 1 + (-8.46 + 3.28i)T + (58.3 - 53.2i)T^{2} \) |
| 83 | \( 1 + (-0.241 + 2.60i)T + (-81.5 - 15.2i)T^{2} \) |
| 89 | \( 1 + (5.56 + 11.1i)T + (-53.6 + 71.0i)T^{2} \) |
| 97 | \( 1 + (-9.64 - 8.78i)T + (8.95 + 96.5i)T^{2} \) |
| 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
−12.16688049623801059860626935643, −11.20614751005571672841995124806, −10.66736223204627725346664030012, −8.789590731387492560474866479559, −7.962340690188298708191717675144, −7.40763901546032624839770932494, −5.82939124186196678229184733869, −4.79618822768254737412444767158, −3.39179301624710546314018191083, −2.66836400689681568071319546265,
0.45697653899144863726378309110, 3.59184905043183919066467563550, 4.47915525818714016716488392799, 4.88905504359937671625349389044, 6.60316865959011744113067184182, 7.31649400639888604395704705933, 8.639857613652105617431494290186, 9.454079026239394227993952069039, 10.84036724778296143470547212586, 11.26332850261053324297339317300
