L(s) = 1 | + (−2.44 − 2.44i)7-s − 6i·11-s + (−2.44 + 2.44i)13-s + (−3.67 + 3.67i)17-s − i·19-s + (3.67 + 3.67i)23-s − 6·29-s − 5·31-s + (4.89 + 4.89i)37-s + 6i·41-s + (7.34 − 7.34i)43-s + (−7.34 + 7.34i)47-s + 4.99i·49-s + (3.67 + 3.67i)53-s + 6·59-s + ⋯ |
L(s) = 1 | + (−0.925 − 0.925i)7-s − 1.80i·11-s + (−0.679 + 0.679i)13-s + (−0.891 + 0.891i)17-s − 0.229i·19-s + (0.766 + 0.766i)23-s − 1.11·29-s − 0.898·31-s + (0.805 + 0.805i)37-s + 0.937i·41-s + (1.12 − 1.12i)43-s + (−1.07 + 1.07i)47-s + 0.714i·49-s + (0.504 + 0.504i)53-s + 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4875839698\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4875839698\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (2.44 + 2.44i)T + 7iT^{2} \) |
| 11 | \( 1 + 6iT - 11T^{2} \) |
| 13 | \( 1 + (2.44 - 2.44i)T - 13iT^{2} \) |
| 17 | \( 1 + (3.67 - 3.67i)T - 17iT^{2} \) |
| 19 | \( 1 + iT - 19T^{2} \) |
| 23 | \( 1 + (-3.67 - 3.67i)T + 23iT^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 5T + 31T^{2} \) |
| 37 | \( 1 + (-4.89 - 4.89i)T + 37iT^{2} \) |
| 41 | \( 1 - 6iT - 41T^{2} \) |
| 43 | \( 1 + (-7.34 + 7.34i)T - 43iT^{2} \) |
| 47 | \( 1 + (7.34 - 7.34i)T - 47iT^{2} \) |
| 53 | \( 1 + (-3.67 - 3.67i)T + 53iT^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 - 5T + 61T^{2} \) |
| 67 | \( 1 + (-4.89 - 4.89i)T + 67iT^{2} \) |
| 71 | \( 1 - 6iT - 71T^{2} \) |
| 73 | \( 1 + (2.44 - 2.44i)T - 73iT^{2} \) |
| 79 | \( 1 + 5iT - 79T^{2} \) |
| 83 | \( 1 + (-11.0 - 11.0i)T + 83iT^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + (4.89 + 4.89i)T + 97iT^{2} \) |
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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
−9.150038136736357771200328189852, −8.342761902387793145762763220270, −7.45354544038000087249186572632, −6.73831842581345718607151098272, −6.11229282156322357748508619849, −5.24963864771479932335618169989, −4.05871428128120696709166372931, −3.55408962817857580446920455499, −2.54464278069317310757072511390, −1.08788171836470897507167675617,
0.17038724921021311031859328030, 2.12375934643402898305572455821, 2.61510987373637880198838515882, 3.79653497241278223998632773181, 4.82885791432710936060983376260, 5.40663223525656386387823545442, 6.42264855240253790929530359760, 7.13174358675713133102103703417, 7.67182266062607312308499091143, 8.875262616979689999950131716527