L(s) = 1 | + 1.73i·2-s − i·3-s − 1.99·4-s + (0.866 + 0.5i)5-s + 1.73·6-s + i·7-s − 1.73i·8-s − 9-s + (−0.866 + 1.49i)10-s − 11-s + 1.99i·12-s + i·13-s − 1.73·14-s + (0.5 − 0.866i)15-s + 0.999·16-s + ⋯ |
L(s) = 1 | + 1.73i·2-s − i·3-s − 1.99·4-s + (0.866 + 0.5i)5-s + 1.73·6-s + i·7-s − 1.73i·8-s − 9-s + (−0.866 + 1.49i)10-s − 11-s + 1.99i·12-s + i·13-s − 1.73·14-s + (0.5 − 0.866i)15-s + 0.999·16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9545113655\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9545113655\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 5 | \( 1 + (-0.866 - 0.5i)T \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 - 1.73iT - T^{2} \) |
| 13 | \( 1 - iT - T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - 1.73T + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 1.73iT - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + iT - T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - 1.73T + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + 1.73iT - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + iT - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 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
−9.850498525654600582808313633287, −9.211986008091777665955213158179, −8.434489391211170350075521359255, −7.64178137068296878299499901789, −6.95256808480429624874253422336, −6.28261202652071340789819072997, −5.50668732918627358443175295474, −5.08220859522155468642935764826, −3.11464624283162556617639222986, −1.94688066311879042378778691487,
0.885842366840783675584110855859, 2.42132954448315411665851392140, 3.29696845915797981696393607920, 4.14974736853597503048552457805, 5.17964953114655331712123517873, 5.59691318252514024965717626508, 7.46376327468835445454077251214, 8.455476660864305045573877659392, 9.424694459457256030869653428988, 9.846554138222408292887525906967