L(s) = 1 | + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (0.5 + 0.866i)5-s − i·7-s + (−0.499 + 0.866i)9-s + (−0.866 + 0.5i)11-s − 0.999·12-s + (0.866 + 0.5i)13-s + (−0.499 + 0.866i)15-s + (−0.499 − 0.866i)16-s + (−0.866 + 0.5i)17-s + (−0.866 + 0.5i)19-s − 0.999·20-s + (0.866 − 0.5i)21-s + 23-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (0.5 + 0.866i)5-s − i·7-s + (−0.499 + 0.866i)9-s + (−0.866 + 0.5i)11-s − 0.999·12-s + (0.866 + 0.5i)13-s + (−0.499 + 0.866i)15-s + (−0.499 − 0.866i)16-s + (−0.866 + 0.5i)17-s + (−0.866 + 0.5i)19-s − 0.999·20-s + (0.866 − 0.5i)21-s + 23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.568 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.568 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.109738735\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.109738735\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 11 | \( 1 + (0.866 - 0.5i)T \) |
| 13 | \( 1 + (-0.866 - 0.5i)T \) |
good | 2 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 + iT - T^{2} \) |
| 17 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 - T + T^{2} \) |
| 29 | \( 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 - iT - T^{2} \) |
| 43 | \( 1 - iT - T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + iT - T^{2} \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 - T + T^{2} \) |
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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
−10.07440924807647504076696835560, −9.447678248947760957390734407606, −8.439373444925856493563416218396, −7.931464426816172393556277995729, −6.95105578370921417301537532694, −6.08234709386471782350709237503, −4.53768958755305388776072007641, −4.25312409411708820449569785589, −3.15516373042862259028956721713, −2.31949389924532411905328967305,
0.935797003416520582094060370667, 2.10257822739947609474732727639, 3.12304263538988759417427620362, 4.80296180888802713390471984093, 5.39577148078529145666952945143, 6.16395899591344341248090368531, 6.98346160776664784460506585596, 8.524884194799226802381186725513, 8.681509100483263139350914431303, 9.135994367734431358108689847200