L(s) = 1 | − 3.82i·2-s − 0.102i·3-s − 10.6·4-s − 0.390·6-s − 9.96·7-s + 25.5i·8-s + 8.98·9-s + 5.29·11-s + 1.08i·12-s + 13.5i·13-s + 38.1i·14-s + 55.1·16-s + 5.23·17-s − 34.4i·18-s + (11.7 + 14.9i)19-s + ⋯ |
L(s) = 1 | − 1.91i·2-s − 0.0340i·3-s − 2.66·4-s − 0.0651·6-s − 1.42·7-s + 3.19i·8-s + 0.998·9-s + 0.481·11-s + 0.0907i·12-s + 1.04i·13-s + 2.72i·14-s + 3.44·16-s + 0.307·17-s − 1.91i·18-s + (0.618 + 0.785i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.618 + 0.785i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.618 + 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9736340728\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9736340728\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 + (-11.7 - 14.9i)T \) |
good | 2 | \( 1 + 3.82iT - 4T^{2} \) |
| 3 | \( 1 + 0.102iT - 9T^{2} \) |
| 7 | \( 1 + 9.96T + 49T^{2} \) |
| 11 | \( 1 - 5.29T + 121T^{2} \) |
| 13 | \( 1 - 13.5iT - 169T^{2} \) |
| 17 | \( 1 - 5.23T + 289T^{2} \) |
| 23 | \( 1 + 37.5T + 529T^{2} \) |
| 29 | \( 1 + 26.4iT - 841T^{2} \) |
| 31 | \( 1 - 29.3iT - 961T^{2} \) |
| 37 | \( 1 - 38.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 39.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 30.3T + 1.84e3T^{2} \) |
| 47 | \( 1 - 77.0T + 2.20e3T^{2} \) |
| 53 | \( 1 + 41.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 50.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 32.2T + 3.72e3T^{2} \) |
| 67 | \( 1 - 100. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 90.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 4.79T + 5.32e3T^{2} \) |
| 79 | \( 1 - 128. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 100.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 127. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 14.5iT - 9.40e3T^{2} \) |
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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
−10.51662254047048015916207632045, −9.821970545471587648967799381782, −9.530552789456329036656465330267, −8.418054329453177443886795483611, −6.99538936243281303639767503900, −5.73068962734827567219335729832, −4.14959040241802321924859167187, −3.75479245779854184740576785535, −2.41188244295441443522766131771, −1.18191777355861772615417622321,
0.48117821697486289719122654130, 3.44714823816537032558015845993, 4.41026110920397651838386315468, 5.67557017390709434539762372690, 6.31156397246385083371342816324, 7.23940084394545636004012289846, 7.81743510150418235940390649571, 9.105300757159450768556059371620, 9.625591824203861859792774093780, 10.40289528257533430492783521442