L(s) = 1 | + (−0.866 − 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s + 0.999·6-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (−0.866 − 0.499i)12-s + (0.366 − 0.366i)13-s + (−0.5 + 0.866i)16-s + (−0.866 + 0.499i)18-s − i·23-s + (0.499 + 0.866i)24-s − i·25-s + (−0.5 + 0.133i)26-s + 0.999i·27-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s + 0.999·6-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (−0.866 − 0.499i)12-s + (0.366 − 0.366i)13-s + (−0.5 + 0.866i)16-s + (−0.866 + 0.499i)18-s − i·23-s + (0.499 + 0.866i)24-s − i·25-s + (−0.5 + 0.133i)26-s + 0.999i·27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5339338826\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5339338826\) |
\(L(1)\) |
|
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 | \( 1 + (0.866 - 0.5i)T \) |
| 23 | \( 1 + iT \) |
good | 5 | \( 1 + iT^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - iT^{2} \) |
| 13 | \( 1 + (-0.366 + 0.366i)T - iT^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - iT^{2} \) |
| 29 | \( 1 + (-0.366 - 0.366i)T + iT^{2} \) |
| 31 | \( 1 - iT - T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 - 1.73T + T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 - T + T^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + (-1 - i)T + iT^{2} \) |
| 61 | \( 1 + iT^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 + 1.73iT - T^{2} \) |
| 73 | \( 1 + iT - T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.37194769638986752996856962616, −9.222396602656160078327937145137, −8.637284979500861446228282475900, −7.59323031575867738679984832486, −6.66773799469694462218209029996, −5.91778301761860912291947015656, −4.66942774344026974642193237903, −3.77565763228613140493452055235, −2.57699348925854865483778556843, −0.928578490718459885996224900777,
1.12099097092326100186523868001, 2.33980725656196735187027177191, 4.17242833116130826884139134974, 5.44400593633297932527337132508, 5.91886927251732852257925010989, 6.92708072689283233344423774153, 7.48798350038660212789109323485, 8.328439170696268085289094074177, 9.336155704489500293183347913577, 9.989277141020695825916440958493