L(s) = 1 | + (−0.5 − 0.866i)2-s + (0.500 − 0.866i)4-s + (−1 + 1.73i)5-s + (0.5 + 0.866i)7-s − 3·8-s + 1.99·10-s + (2 + 3.46i)11-s + (1 − 1.73i)13-s + (0.499 − 0.866i)14-s + (0.500 + 0.866i)16-s + 6·17-s + 4·19-s + (0.999 + 1.73i)20-s + (1.99 − 3.46i)22-s + (0.500 + 0.866i)25-s − 1.99·26-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.250 − 0.433i)4-s + (−0.447 + 0.774i)5-s + (0.188 + 0.327i)7-s − 1.06·8-s + 0.632·10-s + (0.603 + 1.04i)11-s + (0.277 − 0.480i)13-s + (0.133 − 0.231i)14-s + (0.125 + 0.216i)16-s + 1.45·17-s + 0.917·19-s + (0.223 + 0.387i)20-s + (0.426 − 0.738i)22-s + (0.100 + 0.173i)25-s − 0.392·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.25463 - 0.221226i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.25463 - 0.221226i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
good | 2 | \( 1 + (0.5 + 0.866i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2 - 3.46i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1 + 1.73i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1 + 1.73i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + (-1 + 1.73i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + (-6 + 10.3i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + (-8 - 13.8i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (6 + 10.3i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 + (9 + 15.5i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.70731250707372895790968129842, −9.858738689022464400491726851907, −9.338145666460537515354315811243, −7.986067018453158391574882202089, −7.18563224637505461237775186972, −6.16140226457456699023422821152, −5.19548638417327909492650328365, −3.64760841523237647680286086877, −2.67897545623779410770913438795, −1.28787422604450895769164124918,
1.01961052068270479706967825022, 3.13902494119266651398648008825, 4.05831821280693449248314312925, 5.43133796436705680912532916802, 6.34751108222972485962694119457, 7.43596156329780616969054871519, 8.083027083479950121941719978622, 8.828559430282547778411203523050, 9.591201153340732674910822238006, 10.92450201753723108839159723850