L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s + (1.92 + 1.11i)5-s − 0.999·6-s + (2.45 − 0.982i)7-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (−1.11 − 1.92i)10-s + (−0.0571 + 3.31i)11-s + (0.866 + 0.499i)12-s − 0.112·13-s + (−2.61 − 0.377i)14-s + 2.22·15-s + (−0.5 + 0.866i)16-s + (0.119 + 0.207i)17-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.499 − 0.288i)3-s + (0.249 + 0.433i)4-s + (0.861 + 0.497i)5-s − 0.408·6-s + (0.928 − 0.371i)7-s − 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.351 − 0.609i)10-s + (−0.0172 + 0.999i)11-s + (0.249 + 0.144i)12-s − 0.0312·13-s + (−0.699 − 0.100i)14-s + 0.574·15-s + (−0.125 + 0.216i)16-s + (0.0290 + 0.0503i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.283i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.959 + 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.53997 - 0.222653i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.53997 - 0.222653i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.866 + 0.5i)T \) |
| 7 | \( 1 + (-2.45 + 0.982i)T \) |
| 11 | \( 1 + (0.0571 - 3.31i)T \) |
good | 5 | \( 1 + (-1.92 - 1.11i)T + (2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 + 0.112T + 13T^{2} \) |
| 17 | \( 1 + (-0.119 - 0.207i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.218 - 0.377i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.401 - 0.695i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 7.50iT - 29T^{2} \) |
| 31 | \( 1 + (-0.306 + 0.177i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.67 + 6.36i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 3.14T + 41T^{2} \) |
| 43 | \( 1 + 10.1iT - 43T^{2} \) |
| 47 | \( 1 + (11.2 + 6.47i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.28 - 2.22i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.27 + 1.89i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.524 + 0.909i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.48 - 4.29i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 7.58T + 71T^{2} \) |
| 73 | \( 1 + (-2.39 - 4.15i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.429 - 0.247i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 0.569T + 83T^{2} \) |
| 89 | \( 1 + (12.1 + 6.99i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 10.6iT - 97T^{2} \) |
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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
−10.76080195033486513147786933444, −10.12961662188683478321725116254, −9.299153634486488591735479192193, −8.350008046615697249671184083006, −7.41718338079091164568263336021, −6.72443223092521714757548984622, −5.30797886091083104753865942464, −3.95010940282503246150501560039, −2.45587949938776929624967548817, −1.59513444495641555386660291311,
1.42868530370239496974236557087, 2.73247791987475130990420548460, 4.50499403991984032493135047740, 5.52182281531965290422701002615, 6.34217944289783198818799502332, 7.88709267197939988022081412000, 8.344285685704793480240052622531, 9.287885402053347894190973868756, 9.867595947045076637362718060817, 10.99959158848147629626013827672