L(s) = 1 | + (0.866 − 0.5i)2-s + (0.866 + 0.5i)3-s + (0.499 − 0.866i)4-s + (−2.01 + 1.16i)5-s + 0.999·6-s + (1.31 + 2.29i)7-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (−1.16 + 2.01i)10-s + (2.46 + 2.21i)11-s + (0.866 − 0.499i)12-s + 1.44·13-s + (2.28 + 1.32i)14-s − 2.32·15-s + (−0.5 − 0.866i)16-s + (1.60 − 2.78i)17-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (0.499 + 0.288i)3-s + (0.249 − 0.433i)4-s + (−0.901 + 0.520i)5-s + 0.408·6-s + (0.497 + 0.867i)7-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (−0.367 + 0.637i)10-s + (0.743 + 0.668i)11-s + (0.249 − 0.144i)12-s + 0.399·13-s + (0.611 + 0.355i)14-s − 0.600·15-s + (−0.125 − 0.216i)16-s + (0.389 − 0.675i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.905 - 0.425i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.905 - 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.12243 + 0.473465i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.12243 + 0.473465i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.866 - 0.5i)T \) |
| 7 | \( 1 + (-1.31 - 2.29i)T \) |
| 11 | \( 1 + (-2.46 - 2.21i)T \) |
good | 5 | \( 1 + (2.01 - 1.16i)T + (2.5 - 4.33i)T^{2} \) |
| 13 | \( 1 - 1.44T + 13T^{2} \) |
| 17 | \( 1 + (-1.60 + 2.78i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.07 - 5.33i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.14 + 5.44i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 3.26iT - 29T^{2} \) |
| 31 | \( 1 + (-2.67 - 1.54i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.57 + 7.91i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 3.41T + 41T^{2} \) |
| 43 | \( 1 + 1.89iT - 43T^{2} \) |
| 47 | \( 1 + (9.22 - 5.32i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.60 + 7.98i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (0.461 + 0.266i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.233 - 0.404i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.61 + 7.99i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 3.25T + 71T^{2} \) |
| 73 | \( 1 + (-2.50 + 4.33i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.21 + 4.16i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 4.56T + 83T^{2} \) |
| 89 | \( 1 + (3.79 - 2.19i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 17.8iT - 97T^{2} \) |
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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
−11.36279876528378937977869112365, −10.29377861792439090828975679838, −9.438119643118128822904649400233, −8.346201076735388527029828203033, −7.53432480825166404206507333057, −6.38553779158584262290582273630, −5.17985648086171467404978222563, −4.09571897472435468980949918479, −3.25696873925040851708840076829, −1.94226760662342207976657616790,
1.26017861867522612164791279033, 3.35146186828858404039753899816, 4.03497609907603146582589451583, 5.10224159557474015642416504711, 6.45452928472276066060358529603, 7.38274434663981967734049180155, 8.151710697231345059762691731134, 8.796650946620499053980647076503, 10.12133406456081338848633224552, 11.48970688004812187178797936309