L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (2.18 − 3.78i)5-s + (−0.5 − 0.866i)7-s + 0.999·8-s − 4.37·10-s + (−0.686 − 1.18i)11-s + (−1 + 1.73i)13-s + (−0.499 + 0.866i)14-s + (−0.5 − 0.866i)16-s − 1.37·17-s + 5·19-s + (2.18 + 3.78i)20-s + (−0.686 + 1.18i)22-s + (−0.813 + 1.40i)23-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.977 − 1.69i)5-s + (−0.188 − 0.327i)7-s + 0.353·8-s − 1.38·10-s + (−0.206 − 0.358i)11-s + (−0.277 + 0.480i)13-s + (−0.133 + 0.231i)14-s + (−0.125 − 0.216i)16-s − 0.332·17-s + 1.14·19-s + (0.488 + 0.846i)20-s + (−0.146 + 0.253i)22-s + (−0.169 + 0.293i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.547 + 0.836i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.547 + 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.560633 - 1.03733i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.560633 - 1.03733i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.5 + 0.866i)T \) |
good | 5 | \( 1 + (-2.18 + 3.78i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (0.686 + 1.18i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 1.37T + 17T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 + (0.813 - 1.40i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.37 + 7.57i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + (-2.31 + 4.00i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.05 - 7.02i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 8.74T + 53T^{2} \) |
| 59 | \( 1 + (5.05 - 8.76i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.55 + 2.69i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.05 + 1.83i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 7.11T + 71T^{2} \) |
| 73 | \( 1 - 12.1T + 73T^{2} \) |
| 79 | \( 1 + (-2.55 - 4.43i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-8.74 - 15.1i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 14.7T + 89T^{2} \) |
| 97 | \( 1 + (-4.05 - 7.02i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.06040969111084720439223021798, −9.752984480016948679730192480279, −9.482216234507993353007197704572, −8.551501812870936030503557090636, −7.57753157859405492013002111620, −6.01462663131127110654953456431, −5.07219607525224819657031538209, −4.01935908558319669196968515302, −2.24747080160459599453703670913, −0.925166825961432622441268481514,
2.16959692094955805268421014912, 3.33027456357761475730101904344, 5.26204126986445747167639892258, 6.06518795392006075871071646948, 7.00372585369883507184161938496, 7.62179457492301691383139650658, 9.108673482388802673005288463382, 9.845182513350661738337841704416, 10.54741358872363195700114649960, 11.34841912259692096778283861179