L(s) = 1 | + (0.0341 + 0.0592i)2-s + (0.997 − 1.72i)4-s + 2.66·5-s + 0.273·8-s + (0.0910 + 0.157i)10-s + 1.59·11-s + (−2.62 − 4.54i)13-s + (−1.98 − 3.43i)16-s + (−3.27 − 5.67i)17-s + (−0.950 + 1.64i)19-s + (2.65 − 4.60i)20-s + (0.0546 + 0.0946i)22-s + 3.06·23-s + 2.09·25-s + (0.179 − 0.311i)26-s + ⋯ |
L(s) = 1 | + (0.0241 + 0.0418i)2-s + (0.498 − 0.864i)4-s + 1.19·5-s + 0.0965·8-s + (0.0287 + 0.0498i)10-s + 0.482·11-s + (−0.728 − 1.26i)13-s + (−0.496 − 0.859i)16-s + (−0.793 − 1.37i)17-s + (−0.218 + 0.377i)19-s + (0.594 − 1.02i)20-s + (0.0116 + 0.0201i)22-s + 0.639·23-s + 0.419·25-s + (0.0352 − 0.0610i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.214381766\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.214381766\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.0341 - 0.0592i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 - 2.66T + 5T^{2} \) |
| 11 | \( 1 - 1.59T + 11T^{2} \) |
| 13 | \( 1 + (2.62 + 4.54i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (3.27 + 5.67i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.950 - 1.64i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 3.06T + 23T^{2} \) |
| 29 | \( 1 + (-3.19 + 5.53i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (3.35 - 5.81i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.11 - 3.66i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.69 - 6.40i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.63 + 9.75i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.89 - 3.29i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.44 - 7.70i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (5.44 - 9.43i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.35 + 2.35i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.66 + 2.87i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 + (1.09 + 1.90i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.406 + 0.704i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.41 + 5.92i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.235 + 0.407i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.57 + 4.46i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.505094061150116651520282170196, −8.991351345363295438353700811108, −7.64067666769638703835756298312, −6.85698602150961744229457648890, −6.07146033680751374314648428584, −5.38675797142927225388769635621, −4.65087361506827816322225229977, −2.91081006277613962856556477617, −2.16726578472551952058770052626, −0.886418581895892042878480400243,
1.81872521595043251057208803299, 2.38306087400394082719579492408, 3.74575362466332032262654814004, 4.58794892821645242523531308921, 5.82132317717572354552466958663, 6.64626483061798679476609826929, 7.10966937177444880960543923206, 8.298196458247840622803753782186, 9.084145537160382253875135160922, 9.597862247102819918936018414065