L(s) = 1 | + (−0.866 + 0.5i)2-s + (1.44 − 0.948i)3-s + (0.499 − 0.866i)4-s − 0.366·5-s + (−0.780 + 1.54i)6-s + 0.999i·8-s + (1.20 − 2.74i)9-s + (0.317 − 0.183i)10-s − 0.669i·11-s + (−0.0967 − 1.72i)12-s + (0.867 − 0.500i)13-s + (−0.531 + 0.347i)15-s + (−0.5 − 0.866i)16-s + (−2.49 − 4.32i)17-s + (0.334 + 2.98i)18-s + (5.50 + 3.17i)19-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.836 − 0.547i)3-s + (0.249 − 0.433i)4-s − 0.163·5-s + (−0.318 + 0.631i)6-s + 0.353i·8-s + (0.400 − 0.916i)9-s + (0.100 − 0.0579i)10-s − 0.201i·11-s + (−0.0279 − 0.499i)12-s + (0.240 − 0.138i)13-s + (−0.137 + 0.0897i)15-s + (−0.125 − 0.216i)16-s + (−0.605 − 1.04i)17-s + (0.0788 + 0.702i)18-s + (1.26 + 0.729i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.500 + 0.865i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.500 + 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.26177 - 0.727940i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.26177 - 0.727940i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.44 + 0.948i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 0.366T + 5T^{2} \) |
| 11 | \( 1 + 0.669iT - 11T^{2} \) |
| 13 | \( 1 + (-0.867 + 0.500i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (2.49 + 4.32i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-5.50 - 3.17i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 7.69iT - 23T^{2} \) |
| 29 | \( 1 + (-1.58 - 0.914i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-5.47 - 3.16i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.58 + 4.47i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.15 - 3.73i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.24 + 3.89i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (4.16 + 7.21i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4.36 - 7.55i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.29 - 2.47i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.44 + 9.43i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 5.49iT - 71T^{2} \) |
| 73 | \( 1 + (-3.52 + 2.03i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4.17 + 7.23i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (8.50 - 14.7i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-5.35 + 9.27i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-14.9 - 8.60i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.748895170358973216614539543379, −9.029511956290882839961055022897, −8.263257796402014798708647531834, −7.60233215379320556669637083384, −6.79598499999504711197540602506, −5.95288846270542857805898265308, −4.63481557559165536004459240930, −3.32852019332446253529027813140, −2.27453625008116255953639916551, −0.824948009860976761735078269028,
1.56302071255170007141131614629, 2.79270594883393887433676843492, 3.74536690302183263993304472236, 4.65843375647804424415094329619, 5.98311411746212059115629855773, 7.26198668269736992244302759808, 7.935821668188032810598245348435, 8.667999932883394543221887550530, 9.665917708465589978784684955008, 9.853034234368576785904186938859