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 + (0.5 − 0.866i)11-s + (0.866 + 0.499i)12-s + (−0.5 + 0.866i)16-s + i·17-s + (0.866 − 0.499i)18-s − 2·19-s + (0.866 − 0.499i)22-s + (0.499 + 0.866i)24-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 + (0.5 − 0.866i)11-s + (0.866 + 0.499i)12-s + (−0.5 + 0.866i)16-s + i·17-s + (0.866 − 0.499i)18-s − 2·19-s + (0.866 − 0.499i)22-s + (0.499 + 0.866i)24-s − 0.999i·27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 - 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 - 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.447604839\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.447604839\) |
\(L(1)\) |
|
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 | \( 1 + (-0.866 + 0.5i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 - iT - T^{2} \) |
| 19 | \( 1 + 2T + T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 2iT - T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 - T + T^{2} \) |
| 97 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
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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
−9.055366529591435716991555914793, −8.554004507572436256307676986787, −7.976497139364867146572044467996, −6.99913476647046270547739846305, −6.35932534241057453707300763664, −5.71416103453631749697386136369, −4.28594526994680840127388930445, −3.80289844165556543901472994640, −2.76594477286899714820133867788, −1.77935328095653857602908539241,
1.76793121082152067002823293072, 2.56859065300355188838635233570, 3.56601698822310785228126100365, 4.46176799045250495728152669080, 4.87113747219715815344168604260, 6.16901194325622101442255250526, 6.94794582753846297556298730463, 7.79990175788995072187836368129, 8.916934684660653139819644055051, 9.437697900466527160636995367458