| 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
−12.26664675898722879723125588317, −11.44557364329854064145017543180, −10.89688064997335796418235702953, −9.229781392161143327439377342973, −8.245018518645119213245463473597, −7.57727878032114530527210880033, −6.17945872986257115111090914139, −4.65423568644921511740676293150, −3.70326516729052356650578668524, −2.53354513126967766156119519342,
0.930147933845549804229175208277, 3.64108175830880532467656655183, 4.54849542664153722510807865347, 5.60835540233962332377433746898, 6.99922479368061964491743494678, 7.73783693639561184034104661546, 8.931615354144748414358797308164, 10.13433498266989367178984732028, 11.00709451285926415698059333703, 11.87977939278536059869681914602
