| L(s) = 1 | + (0.614 + 0.515i)2-s + (−0.235 − 1.33i)4-s + (−2.58 − 0.940i)5-s + (0.412 − 2.34i)7-s + (1.34 − 2.33i)8-s + (−1.10 − 1.90i)10-s + (−0.235 + 0.0855i)11-s + (2.00 − 1.67i)13-s + (1.45 − 1.22i)14-s + (−0.524 + 0.190i)16-s + (−0.146 − 0.254i)17-s + (1.39 − 2.41i)19-s + (−0.648 + 3.67i)20-s + (−0.188 − 0.0685i)22-s + (1.16 + 6.58i)23-s + ⋯ |
| L(s) = 1 | + (0.434 + 0.364i)2-s + (−0.117 − 0.668i)4-s + (−1.15 − 0.420i)5-s + (0.155 − 0.884i)7-s + (0.475 − 0.823i)8-s + (−0.348 − 0.603i)10-s + (−0.0708 + 0.0257i)11-s + (0.554 − 0.465i)13-s + (0.390 − 0.327i)14-s + (−0.131 + 0.0477i)16-s + (−0.0355 − 0.0616i)17-s + (0.319 − 0.553i)19-s + (−0.144 + 0.822i)20-s + (−0.0401 − 0.0146i)22-s + (0.242 + 1.37i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.334 + 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.334 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.991236 - 0.699828i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.991236 - 0.699828i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (-0.614 - 0.515i)T + (0.347 + 1.96i)T^{2} \) |
| 5 | \( 1 + (2.58 + 0.940i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.412 + 2.34i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (0.235 - 0.0855i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-2.00 + 1.67i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.146 + 0.254i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.39 + 2.41i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.16 - 6.58i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-0.271 - 0.228i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.480 - 2.72i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-3.49 - 6.05i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-7.44 + 6.24i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (0.244 - 0.0891i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.98 + 11.2i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 5.43T + 53T^{2} \) |
| 59 | \( 1 + (-5.61 - 2.04i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (2.05 - 11.6i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-1.38 + 1.16i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (0.185 + 0.320i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.51 - 4.35i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.614 - 0.516i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-2.11 - 1.77i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-5.22 + 9.05i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-13.9 + 5.07i)T + (74.3 - 62.3i)T^{2} \) |
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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
−11.87977939278536059869681914602, −11.00709451285926415698059333703, −10.13433498266989367178984732028, −8.931615354144748414358797308164, −7.73783693639561184034104661546, −6.99922479368061964491743494678, −5.60835540233962332377433746898, −4.54849542664153722510807865347, −3.64108175830880532467656655183, −0.930147933845549804229175208277,
2.53354513126967766156119519342, 3.70326516729052356650578668524, 4.65423568644921511740676293150, 6.17945872986257115111090914139, 7.57727878032114530527210880033, 8.245018518645119213245463473597, 9.229781392161143327439377342973, 10.89688064997335796418235702953, 11.44557364329854064145017543180, 12.26664675898722879723125588317
