L(s) = 1 | + (−0.850 − 0.490i)2-s + (0.900 + 1.47i)3-s + (−0.518 − 0.897i)4-s + (−0.940 − 1.62i)5-s + (−0.0387 − 1.69i)6-s + 2.98i·8-s + (−1.37 + 2.66i)9-s + 1.84i·10-s + (3.54 + 2.04i)11-s + (0.861 − 1.57i)12-s + (3.51 − 2.02i)13-s + (1.56 − 2.85i)15-s + (0.426 − 0.738i)16-s + 1.62·17-s + (2.48 − 1.58i)18-s − 8.12i·19-s + ⋯ |
L(s) = 1 | + (−0.601 − 0.347i)2-s + (0.519 + 0.854i)3-s + (−0.259 − 0.448i)4-s + (−0.420 − 0.728i)5-s + (−0.0158 − 0.693i)6-s + 1.05i·8-s + (−0.459 + 0.887i)9-s + 0.583i·10-s + (1.06 + 0.616i)11-s + (0.248 − 0.454i)12-s + (0.974 − 0.562i)13-s + (0.403 − 0.737i)15-s + (0.106 − 0.184i)16-s + 0.393·17-s + (0.584 − 0.374i)18-s − 1.86i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.852 + 0.522i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.852 + 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.06838 - 0.301653i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.06838 - 0.301653i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.900 - 1.47i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.850 + 0.490i)T + (1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (0.940 + 1.62i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-3.54 - 2.04i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.51 + 2.02i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 1.62T + 17T^{2} \) |
| 19 | \( 1 + 8.12iT - 19T^{2} \) |
| 23 | \( 1 + (-3.73 + 2.15i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.542 + 0.313i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.69 + 2.13i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 7.94T + 37T^{2} \) |
| 41 | \( 1 + (-0.912 - 1.57i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.53 - 6.12i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (3.96 - 6.87i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 8.37iT - 53T^{2} \) |
| 59 | \( 1 + (4.08 + 7.07i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.24 - 1.87i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.26 + 10.8i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 14.4iT - 71T^{2} \) |
| 73 | \( 1 + 3.78iT - 73T^{2} \) |
| 79 | \( 1 + (4.18 - 7.24i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.38 + 7.59i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 9.80T + 89T^{2} \) |
| 97 | \( 1 + (-11.4 - 6.61i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.00487038742515286080215482925, −9.922495872683639860346890655715, −9.218199820303653720434434036109, −8.699499784106664194063688330827, −7.84505001829921718402586710378, −6.25495114660464249029682495462, −4.88841818178464203023199385163, −4.35193526735934922499442641846, −2.81638955273931277061770565594, −1.04482916875103606062861481024,
1.30922053236643029064206669358, 3.33208354362497151930277380990, 3.81048076586861454492218812797, 6.05198630490452610461195464488, 6.79991766123867583458611905180, 7.59973209902777476081388366761, 8.437061337837664554536560706437, 9.019537532045769672056996198584, 10.08168753662713931354046290203, 11.41548085324767343673096521783