L(s) = 1 | + (0.866 + 0.5i)2-s + (−0.5 − 0.866i)3-s + (0.499 + 0.866i)4-s + (3.60 + 2.08i)5-s − 0.999i·6-s + (2.59 + 0.5i)7-s + 0.999i·8-s + (−0.499 + 0.866i)9-s + (2.08 + 3.60i)10-s + (−1.87 + 1.08i)11-s + (0.499 − 0.866i)12-s + (−3.58 − 0.418i)13-s + (2 + 1.73i)14-s − 4.16i·15-s + (−0.5 + 0.866i)16-s + (0.581 + 1.00i)17-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (−0.288 − 0.499i)3-s + (0.249 + 0.433i)4-s + (1.61 + 0.930i)5-s − 0.408i·6-s + (0.981 + 0.188i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.658 + 1.13i)10-s + (−0.564 + 0.325i)11-s + (0.144 − 0.249i)12-s + (−0.993 − 0.116i)13-s + (0.534 + 0.462i)14-s − 1.07i·15-s + (−0.125 + 0.216i)16-s + (0.140 + 0.244i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.693 - 0.720i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.693 - 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.23662 + 0.950924i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.23662 + 0.950924i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (-2.59 - 0.5i)T \) |
| 13 | \( 1 + (3.58 + 0.418i)T \) |
good | 5 | \( 1 + (-3.60 - 2.08i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1.87 - 1.08i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-0.581 - 1.00i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (5.47 + 3.16i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.58 + 4.47i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 2.16T + 29T^{2} \) |
| 31 | \( 1 + (-7.79 + 4.5i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (5.47 + 3.16i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 2.83T + 43T^{2} \) |
| 47 | \( 1 + (-4.75 - 2.74i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.0811 + 0.140i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-7.06 + 4.08i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.74 + 3.01i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (7.93 - 4.58i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 16.3iT - 71T^{2} \) |
| 73 | \( 1 + (-5.47 + 3.16i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4.5 - 7.79i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 14.8iT - 83T^{2} \) |
| 89 | \( 1 + (-2.73 - 1.58i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 11iT - 97T^{2} \) |
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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
−10.78215650015069432710175503806, −10.36137996434422511237197854960, −9.132617198017579004954402995810, −7.984525047055138902321400261704, −7.01647077950114823068900036816, −6.33913359375978612908192992346, −5.41832720398689066613883616193, −4.67562448903059900326033675087, −2.62305506636265900454286720588, −2.07548143501399806461155647575,
1.42519533479334335192575807059, 2.54986259097196845332176670547, 4.30253358186211321360230276629, 5.19429499520556748014086681074, 5.53087207796110772343971917507, 6.75221356091468336050711498312, 8.240989522255113199657312556476, 9.109705958283667905408957815233, 10.13980123798430372390980094983, 10.44283161591140078083035582755