L(s) = 1 | + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (1.40 − 1.40i)5-s + (−0.707 + 0.707i)8-s + (0.5 + 0.866i)9-s + (−0.991 + 1.71i)10-s + (0.793 − 0.608i)13-s + (0.500 − 0.866i)16-s + (−0.793 − 1.37i)17-s + (−0.707 − 0.707i)18-s + (0.513 − 1.91i)20-s − 2.93i·25-s + (−0.608 + 0.793i)26-s + (−0.258 + 0.448i)29-s + (−0.258 + 0.965i)32-s + ⋯ |
L(s) = 1 | + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (1.40 − 1.40i)5-s + (−0.707 + 0.707i)8-s + (0.5 + 0.866i)9-s + (−0.991 + 1.71i)10-s + (0.793 − 0.608i)13-s + (0.500 − 0.866i)16-s + (−0.793 − 1.37i)17-s + (−0.707 − 0.707i)18-s + (0.513 − 1.91i)20-s − 2.93i·25-s + (−0.608 + 0.793i)26-s + (−0.258 + 0.448i)29-s + (−0.258 + 0.965i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.704 + 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.704 + 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.072319022\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.072319022\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.965 - 0.258i)T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + (-0.793 + 0.608i)T \) |
good | 3 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + (-1.40 + 1.40i)T - iT^{2} \) |
| 11 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 + (0.793 + 1.37i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.258 - 0.448i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + iT^{2} \) |
| 37 | \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 41 | \( 1 + (-0.315 - 1.17i)T + (-0.866 + 0.5i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + 1.73T + T^{2} \) |
| 59 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 61 | \( 1 + (0.226 - 0.130i)T + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 73 | \( 1 + (0.184 + 0.184i)T + iT^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + (-0.739 + 0.198i)T + (0.866 - 0.5i)T^{2} \) |
| 97 | \( 1 + (-1.78 - 0.478i)T + (0.866 + 0.5i)T^{2} \) |
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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
−9.089185756883209852669066752037, −8.335690508679993917767642967074, −7.73301028446678426422387194164, −6.64223645154631911059244991818, −5.98880361813999010392216315957, −5.12396816349335680027840528405, −4.67793885216944611727402469825, −2.81267050521210870004545256988, −1.86273112302379065229085766145, −1.05268000100951820396998124683,
1.58110258221833642347405152808, 2.19158438885807402735469869080, 3.29508951790340169214649775859, 4.02500168328634342154286829043, 5.85610681732794255070096630256, 6.29262047081334941709325674388, 6.80708687809908268750517549098, 7.57072175180945436299149423893, 8.711557673535153587808703538691, 9.315322989783447885798933987244