L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s + (1.99 − 1.01i)5-s + (−0.499 − 0.866i)6-s − 2.79i·7-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (−2.23 − 0.118i)10-s + 4.02·11-s + 0.999i·12-s + (−0.0960 + 0.0554i)13-s + (−1.39 + 2.42i)14-s + (2.23 + 0.118i)15-s + (−0.5 + 0.866i)16-s + (−3.68 − 2.12i)17-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.499 + 0.288i)3-s + (0.249 + 0.433i)4-s + (0.891 − 0.453i)5-s + (−0.204 − 0.353i)6-s − 1.05i·7-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (−0.706 − 0.0375i)10-s + 1.21·11-s + 0.288i·12-s + (−0.0266 + 0.0153i)13-s + (−0.373 + 0.647i)14-s + (0.576 + 0.0306i)15-s + (−0.125 + 0.216i)16-s + (−0.893 − 0.515i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.601 + 0.798i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.601 + 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.37169 - 0.683958i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.37169 - 0.683958i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.866 - 0.5i)T \) |
| 5 | \( 1 + (-1.99 + 1.01i)T \) |
| 19 | \( 1 + (-0.163 + 4.35i)T \) |
good | 7 | \( 1 + 2.79iT - 7T^{2} \) |
| 11 | \( 1 - 4.02T + 11T^{2} \) |
| 13 | \( 1 + (0.0960 - 0.0554i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (3.68 + 2.12i)T + (8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (7.65 - 4.42i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.907 - 1.57i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 7.88T + 31T^{2} \) |
| 37 | \( 1 + 1.68iT - 37T^{2} \) |
| 41 | \( 1 + (3.88 - 6.72i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-6.46 - 3.73i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.17 + 2.40i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.45 + 3.72i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (0.188 - 0.326i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (6.18 + 10.7i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.82 + 2.21i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (5.90 - 10.2i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (8.33 + 4.81i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.11 - 3.66i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 8.25iT - 83T^{2} \) |
| 89 | \( 1 + (-5.01 - 8.68i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.52 - 2.61i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.35608777481000297347762994316, −9.641858078395986980651429250683, −9.108577017456425461207298259463, −8.196448956726481012207947876506, −7.09546994484162036541412400594, −6.28020879469172977733939000918, −4.71299773661354771548814764568, −3.85771312689703385980834570457, −2.41973154541398015910941192412, −1.13223134423734512717517007884,
1.71735132474871084527421014886, 2.58348473370696086627585121706, 4.19017264378672729251127410006, 5.96495399342482724133741744789, 6.18353920235418424916159857756, 7.28299156995891424411407185303, 8.552795767443024726514801729222, 8.877021019248485771385311164528, 9.889555539581620776230901424433, 10.50764542131977147596850381842