L(s) = 1 | + (−0.866 + 0.5i)2-s + 1.73·3-s + (−0.500 + 0.866i)4-s + (1 + 2i)5-s + (−1.49 + 0.866i)6-s + (0.866 − 2.5i)7-s − 3i·8-s + 2.99·9-s + (−1.86 − 1.23i)10-s + 6·11-s + (−0.866 + 1.49i)12-s + (−3.46 + 2i)13-s + (0.500 + 2.59i)14-s + (1.73 + 3.46i)15-s + (0.500 + 0.866i)16-s + (−1.73 + i)17-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + 1.00·3-s + (−0.250 + 0.433i)4-s + (0.447 + 0.894i)5-s + (−0.612 + 0.353i)6-s + (0.327 − 0.944i)7-s − 1.06i·8-s + 0.999·9-s + (−0.590 − 0.389i)10-s + 1.80·11-s + (−0.250 + 0.433i)12-s + (−0.960 + 0.554i)13-s + (0.133 + 0.694i)14-s + (0.447 + 0.894i)15-s + (0.125 + 0.216i)16-s + (−0.420 + 0.242i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.483 - 0.875i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.483 - 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.20648 + 0.711947i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.20648 + 0.711947i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 1.73T \) |
| 5 | \( 1 + (-1 - 2i)T \) |
| 7 | \( 1 + (-0.866 + 2.5i)T \) |
good | 2 | \( 1 + (0.866 - 0.5i)T + (1 - 1.73i)T^{2} \) |
| 11 | \( 1 - 6T + 11T^{2} \) |
| 13 | \( 1 + (3.46 - 2i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (1.73 - i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3 - 5.19i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 3iT - 23T^{2} \) |
| 29 | \( 1 + (1 - 1.73i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (6.92 + 4i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1 + 1.73i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.866 + 0.5i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (2.59 - 1.5i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-10.3 + 6i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.5 + 6.06i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.06 - 3.5i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (7 + 12.1i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (3.46 + 2i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (1.5 - 2.59i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.73 - i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.83966487126700602724876645978, −10.47028106353820368393913852220, −9.765238087185316169222789417328, −8.987824599706140055202899610739, −8.050574041233869949281626929797, −7.01729054300237932491353457685, −6.64617970890918861813327199336, −4.23702119251192064689376742855, −3.62533051785185703990523860286, −1.86224008991233329012157758872,
1.40762383048239847299199665577, 2.49441936939358577103692945835, 4.40442794404849228821731066920, 5.33189844135013584119812763553, 6.74123410997559027340348339404, 8.263940500713275156923381725042, 8.958428636738844619955239356839, 9.311405639516531243880197818431, 10.16724629522007198385099601818, 11.55981619489762695797830304446