L(s) = 1 | + (−1.30 + 2.26i)2-s + (−2.42 − 4.20i)4-s + (1.11 − 1.93i)5-s + (−2 − 1.73i)7-s + 7.47·8-s + (2.92 + 5.06i)10-s + (−1.5 − 2.59i)11-s − 13-s + (6.54 − 2.26i)14-s + (−4.92 + 8.53i)16-s + (0.736 + 1.27i)17-s + (−1.5 + 2.59i)19-s − 10.8·20-s + 7.85·22-s + (−4.11 + 7.13i)23-s + ⋯ |
L(s) = 1 | + (−0.925 + 1.60i)2-s + (−1.21 − 2.10i)4-s + (0.499 − 0.866i)5-s + (−0.755 − 0.654i)7-s + 2.64·8-s + (0.925 + 1.60i)10-s + (−0.452 − 0.783i)11-s − 0.277·13-s + (1.74 − 0.605i)14-s + (−1.23 + 2.13i)16-s + (0.178 + 0.309i)17-s + (−0.344 + 0.596i)19-s − 2.42·20-s + 1.67·22-s + (−0.858 + 1.48i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (2 + 1.73i)T \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 + (1.30 - 2.26i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (-1.11 + 1.93i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-0.736 - 1.27i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.5 - 2.59i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.11 - 7.13i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 + (2.5 + 4.33i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.35 - 4.07i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 4.47T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + (3.73 - 6.47i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3.73 + 6.47i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (0.736 + 1.27i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.5 - 2.59i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.5 - 2.59i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 8.94T + 71T^{2} \) |
| 73 | \( 1 + (-1.35 - 2.34i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.35 + 2.34i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (-1.11 + 1.93i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 9.41T + 97T^{2} \) |
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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
−9.685189091288742995533559344134, −9.022596016919218622266460582322, −8.071750805877635202654697591420, −7.55980208973122466636735299024, −6.44397900283482383999943819474, −5.78329784947620868191492817450, −5.07413648051854586608028194205, −3.72307114132072439895375022139, −1.44753318114864922335316415897, 0,
2.10763991469750759528048455278, 2.60344181598290445483504774027, 3.61803831127645157847767493753, 4.92359572477605806208166223835, 6.38106465960017753553064942877, 7.29600919162277366229111860774, 8.379930000021349043469553804432, 9.213795769930802814763199912084, 9.876489472587370965440491531903