L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (−0.5 − 0.866i)5-s − i·7-s − 0.999·8-s + (0.499 − 0.866i)10-s + (−0.866 + 1.5i)11-s − 1.73·13-s + (0.866 − 0.5i)14-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)19-s + 0.999·20-s − 1.73·22-s + (0.5 + 0.866i)23-s + (−0.499 + 0.866i)25-s + (−0.866 − 1.49i)26-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (−0.5 − 0.866i)5-s − i·7-s − 0.999·8-s + (0.499 − 0.866i)10-s + (−0.866 + 1.5i)11-s − 1.73·13-s + (0.866 − 0.5i)14-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)19-s + 0.999·20-s − 1.73·22-s + (0.5 + 0.866i)23-s + (−0.499 + 0.866i)25-s + (−0.866 − 1.49i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.08446193011\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.08446193011\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + iT \) |
good | 11 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + 1.73T + T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + 1.73T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 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 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 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
−8.652853040593289328294401865347, −7.73591576737450679451602010157, −7.26572052915175137090985305048, −6.91934129130575608049375510179, −5.38543127815873408457543911654, −4.84412260915764077913636537256, −4.43107409423867223107507668463, −3.38632115651785794008771527749, −2.10220225959405533417816058198, −0.04275398593791337320123136396,
2.04725755395559543950119135283, 2.92491642751667109253475955095, 3.30212153460984912665087715354, 4.65011811867723150749378010539, 5.29499869684488890435674153198, 6.13634965895390290746460696705, 6.86255026167303727379833507363, 8.153846594440131419274794241338, 8.473068151362092574178137249243, 9.614682354268311251023725335910