L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.707 − 0.707i)3-s + (0.499 + 0.866i)4-s + (−0.707 − 0.707i)5-s + (−0.965 + 0.258i)6-s + (0.866 + 0.5i)7-s − 0.999i·8-s − 1.00i·9-s + (0.258 + 0.965i)10-s + (0.965 + 0.258i)12-s + (1.22 − 0.707i)13-s + (−0.499 − 0.866i)14-s − 1.00·15-s + (−0.5 + 0.866i)16-s + (−0.500 + 0.866i)18-s − 1.93·19-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.707 − 0.707i)3-s + (0.499 + 0.866i)4-s + (−0.707 − 0.707i)5-s + (−0.965 + 0.258i)6-s + (0.866 + 0.5i)7-s − 0.999i·8-s − 1.00i·9-s + (0.258 + 0.965i)10-s + (0.965 + 0.258i)12-s + (1.22 − 0.707i)13-s + (−0.499 − 0.866i)14-s − 1.00·15-s + (−0.5 + 0.866i)16-s + (−0.500 + 0.866i)18-s − 1.93·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.422 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.422 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.001982685\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.001982685\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (0.707 + 0.707i)T \) |
| 7 | \( 1 + (-0.866 - 0.5i)T \) |
good | 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + 1.93T + T^{2} \) |
| 23 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.965 + 1.67i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + 1.73T + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
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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
−8.648755424438601491229444783724, −8.391542761708705346115184144759, −7.70953856097570732273700320218, −6.85113408986696752919164470525, −5.98183901851900435244832990985, −4.64146840707818720527583581705, −3.79322879956393887993375213166, −2.83635396406885614419112803069, −1.84063468503155436577055106439, −0.885266853027334469366328742467,
1.53502837095954838718684145697, 2.62288865317525875937267178256, 3.83119141101972613898093414660, 4.43583001645240909543863249717, 5.48141947192830191665642279098, 6.63757282720938611942287970890, 7.16245831403649169969701411895, 8.056061082243347406341869675034, 8.518152216688921670251984171300, 9.060827801694626625253217697607