L(s) = 1 | + (−0.965 − 0.258i)2-s + (0.923 + 0.382i)3-s + (0.866 + 0.499i)4-s + (−0.382 − 0.923i)5-s + (−0.793 − 0.608i)6-s + (0.258 − 0.965i)7-s + (−0.707 − 0.707i)8-s + (0.707 + 0.707i)9-s + (0.130 + 0.991i)10-s + (0.608 + 0.793i)12-s + (−0.478 − 1.78i)13-s + (−0.499 + 0.866i)14-s − i·15-s + (0.500 + 0.866i)16-s + (−0.5 − 0.866i)18-s − 1.21i·19-s + ⋯ |
L(s) = 1 | + (−0.965 − 0.258i)2-s + (0.923 + 0.382i)3-s + (0.866 + 0.499i)4-s + (−0.382 − 0.923i)5-s + (−0.793 − 0.608i)6-s + (0.258 − 0.965i)7-s + (−0.707 − 0.707i)8-s + (0.707 + 0.707i)9-s + (0.130 + 0.991i)10-s + (0.608 + 0.793i)12-s + (−0.478 − 1.78i)13-s + (−0.499 + 0.866i)14-s − i·15-s + (0.500 + 0.866i)16-s + (−0.5 − 0.866i)18-s − 1.21i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9022653273\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9022653273\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.965 + 0.258i)T \) |
| 3 | \( 1 + (-0.923 - 0.382i)T \) |
| 5 | \( 1 + (0.382 + 0.923i)T \) |
| 7 | \( 1 + (-0.258 + 0.965i)T \) |
good | 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.478 + 1.78i)T + (-0.866 + 0.5i)T^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 + 1.21iT - T^{2} \) |
| 23 | \( 1 + (1.86 - 0.5i)T + (0.866 - 0.5i)T^{2} \) |
| 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 | \( 1 + (-0.382 + 0.662i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.793 - 1.37i)T + (-0.5 + 0.866i)T^{2} \) |
| 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 | \( 1 + (-0.198 + 0.739i)T + (-0.866 - 0.5i)T^{2} \) |
| 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
−8.712579853832317743904472596941, −8.254639759188132589060139596622, −7.62248252056786100134855633790, −7.18004122766795120399828868107, −5.73508813926161005244589581562, −4.70859183282133540277261758520, −3.87078357187612953655538016652, −3.07246450541036422496957855148, −1.95874048723250395248353541971, −0.69847549036653746355291183435,
1.95276821965764343499660290704, 2.18394910971531325437206815374, 3.39032927909670603293622582854, 4.38406118817166593723343001283, 5.90075112129201158524856949019, 6.48625100534317922264659178742, 7.17375098781428113771384522187, 7.979154575521208145703614109322, 8.377056035389600240216522009377, 9.255413009970473077531949231762