L(s) = 1 | + (0.5 + 0.866i)3-s + (0.866 + 0.5i)5-s + (0.866 − 0.5i)7-s + (−0.499 + 0.866i)9-s + (−1.5 + 0.866i)11-s + 0.999i·15-s + (0.866 + 0.499i)21-s − 0.999·27-s + 1.73·29-s + (−0.866 − 1.5i)31-s + (−1.5 − 0.866i)33-s + 0.999·35-s + (−0.866 + 0.499i)45-s + (0.499 − 0.866i)49-s + (−0.866 − 1.5i)53-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)3-s + (0.866 + 0.5i)5-s + (0.866 − 0.5i)7-s + (−0.499 + 0.866i)9-s + (−1.5 + 0.866i)11-s + 0.999i·15-s + (0.866 + 0.499i)21-s − 0.999·27-s + 1.73·29-s + (−0.866 − 1.5i)31-s + (−1.5 − 0.866i)33-s + 0.999·35-s + (−0.866 + 0.499i)45-s + (0.499 − 0.866i)49-s + (−0.866 − 1.5i)53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.378 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.378 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.447964396\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.447964396\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (-0.866 + 0.5i)T \) |
good | 5 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 - 1.73T + T^{2} \) |
| 31 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - T + T^{2} \) |
| 89 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 - 1.73iT - T^{2} \) |
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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
−10.03014076204099189335427300800, −9.399619969219002912617044879771, −8.198365700856281798314647197309, −7.80487893253625665380189767429, −6.74323885533229220343070846103, −5.52712815536586865112860961791, −4.91281096068196826600669154374, −4.04556870434373230354377839356, −2.71064254738686753267134994663, −2.05498333434485971264513030392,
1.30943084078403990086472268939, 2.33876627395982645264829538381, 3.17304829934716906974248437885, 4.86607239995514020870196075598, 5.51452856963098921594943686419, 6.26562205379882099368558365486, 7.38495220948860389074848444487, 8.218948284176846137266759215836, 8.623749198175470930078247888345, 9.431582520513879569095754467496