L(s) = 1 | + (0.258 + 0.965i)2-s + (0.923 − 0.382i)3-s + (−0.866 + 0.499i)4-s + (0.130 − 0.991i)5-s + (0.608 + 0.793i)6-s + (0.965 − 0.258i)7-s + (−0.707 − 0.707i)8-s + (0.707 − 0.707i)9-s + (0.991 − 0.130i)10-s + (−0.608 + 0.793i)12-s + (−0.739 − 0.198i)13-s + (0.499 + 0.866i)14-s + (−0.258 − 0.965i)15-s + (0.500 − 0.866i)16-s + (0.866 + 0.5i)18-s − 1.58i·19-s + ⋯ |
L(s) = 1 | + (0.258 + 0.965i)2-s + (0.923 − 0.382i)3-s + (−0.866 + 0.499i)4-s + (0.130 − 0.991i)5-s + (0.608 + 0.793i)6-s + (0.965 − 0.258i)7-s + (−0.707 − 0.707i)8-s + (0.707 − 0.707i)9-s + (0.991 − 0.130i)10-s + (−0.608 + 0.793i)12-s + (−0.739 − 0.198i)13-s + (0.499 + 0.866i)14-s + (−0.258 − 0.965i)15-s + (0.500 − 0.866i)16-s + (0.866 + 0.5i)18-s − 1.58i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0136i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0136i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.852219492\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.852219492\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.258 - 0.965i)T \) |
| 3 | \( 1 + (-0.923 + 0.382i)T \) |
| 5 | \( 1 + (-0.130 + 0.991i)T \) |
| 7 | \( 1 + (-0.965 + 0.258i)T \) |
good | 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.739 + 0.198i)T + (0.866 + 0.5i)T^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 + 1.58iT - T^{2} \) |
| 23 | \( 1 + (0.5 - 1.86i)T + (-0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 47 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + (-0.923 - 1.60i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.608 - 1.05i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 - 0.517iT - T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + (0.448 + 0.258i)T + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-1.78 + 0.478i)T + (0.866 - 0.5i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
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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
−8.909981474495522561365333094234, −8.245244390125979768463276818877, −7.48600272603008584266797254452, −7.20395781312772222890240278809, −5.94915916938820277294130598244, −5.09723664344136126087228579377, −4.50839343050454304091290382873, −3.64979120012138503977658641896, −2.39481762225651305885779140614, −1.11394163894436973670651784146,
1.84421288783975281463581377949, 2.34281226271393057627217924143, 3.28904626276815583210406092636, 4.12981512052399918942784620199, 4.83445545121273301895724783376, 5.79890613541236349964134852231, 6.83506877493168459051074466674, 8.047635034003965538328140472821, 8.220946423956538102604283850832, 9.345982457771090718730497448024