L(s) = 1 | + (1.19 − 2.06i)2-s + (−1.34 − 1.08i)3-s + (−1.84 − 3.20i)4-s + (−1.46 − 2.52i)5-s + (−3.85 + 1.49i)6-s − 4.05·8-s + (0.637 + 2.93i)9-s − 6.97·10-s + (0.676 − 1.17i)11-s + (−0.987 + 6.32i)12-s + (0.733 + 1.26i)13-s + (−0.779 + 4.99i)15-s + (−1.13 + 1.96i)16-s + 3.31·17-s + (6.82 + 2.18i)18-s − 2.20·19-s + ⋯ |
L(s) = 1 | + (0.843 − 1.46i)2-s + (−0.778 − 0.627i)3-s + (−0.924 − 1.60i)4-s + (−0.653 − 1.13i)5-s + (−1.57 + 0.608i)6-s − 1.43·8-s + (0.212 + 0.977i)9-s − 2.20·10-s + (0.204 − 0.353i)11-s + (−0.284 + 1.82i)12-s + (0.203 + 0.352i)13-s + (−0.201 + 1.29i)15-s + (−0.284 + 0.492i)16-s + 0.802·17-s + (1.60 + 0.514i)18-s − 0.506·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.533 - 0.845i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.533 - 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.602978 + 1.09375i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.602978 + 1.09375i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.34 + 1.08i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-1.19 + 2.06i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (1.46 + 2.52i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.676 + 1.17i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.733 - 1.26i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 3.31T + 17T^{2} \) |
| 19 | \( 1 + 2.20T + 19T^{2} \) |
| 23 | \( 1 + (1.31 + 2.27i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.521 + 0.903i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.63 - 2.83i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 41 | \( 1 + (-0.904 - 1.56i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.17 - 3.76i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.98 + 3.44i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 6.45T + 53T^{2} \) |
| 59 | \( 1 + (6.10 + 10.5i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.279 + 0.484i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.40 + 11.0i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 12.9T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 + (0.383 - 0.664i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.983 + 1.70i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 6.40T + 89T^{2} \) |
| 97 | \( 1 + (-4.14 + 7.17i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.90976832414913408482272454931, −10.08268629103618180451313034379, −8.832245708751951561358042530727, −7.88382451393826660369771637269, −6.44326981167731814017139311174, −5.27413228531455941169837461247, −4.61525112765182431446918877846, −3.54152298639352697267958007160, −1.86509311907150837187415458838, −0.69268386286219370261094706363,
3.37831316458771277531297714115, 4.10096284721631459774211876902, 5.23998408679832331047864623766, 6.07401024717069481969169658391, 6.93151182948344019294045198062, 7.55928098599017941844376470604, 8.715155576009834591145585379830, 10.08711330824915765629522928689, 10.82590794373649655778197842199, 11.87055314109708894295715350268