L(s) = 1 | + (−0.809 + 2.48i)2-s + (0.809 − 0.587i)3-s + (−3.92 − 2.85i)4-s + (−0.190 − 0.587i)5-s + (0.809 + 2.48i)6-s + (0.809 + 0.587i)7-s + (6.04 − 4.39i)8-s + (0.309 − 0.951i)9-s + 1.61·10-s + (−3.30 − 0.224i)11-s − 4.85·12-s + (0.0729 − 0.224i)13-s + (−2.11 + 1.53i)14-s + (−0.5 − 0.363i)15-s + (3.04 + 9.37i)16-s + (−0.354 − 1.08i)17-s + ⋯ |
L(s) = 1 | + (−0.572 + 1.76i)2-s + (0.467 − 0.339i)3-s + (−1.96 − 1.42i)4-s + (−0.0854 − 0.262i)5-s + (0.330 + 1.01i)6-s + (0.305 + 0.222i)7-s + (2.13 − 1.55i)8-s + (0.103 − 0.317i)9-s + 0.511·10-s + (−0.997 − 0.0676i)11-s − 1.40·12-s + (0.0202 − 0.0622i)13-s + (−0.566 + 0.411i)14-s + (−0.129 − 0.0937i)15-s + (0.761 + 2.34i)16-s + (−0.0858 − 0.264i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0219 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0219 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.417216 + 0.408142i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.417216 + 0.408142i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.809 + 0.587i)T \) |
| 11 | \( 1 + (3.30 + 0.224i)T \) |
good | 2 | \( 1 + (0.809 - 2.48i)T + (-1.61 - 1.17i)T^{2} \) |
| 5 | \( 1 + (0.190 + 0.587i)T + (-4.04 + 2.93i)T^{2} \) |
| 7 | \( 1 + (-0.809 - 0.587i)T + (2.16 + 6.65i)T^{2} \) |
| 13 | \( 1 + (-0.0729 + 0.224i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (0.354 + 1.08i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (4.73 - 3.44i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 - 0.236T + 23T^{2} \) |
| 29 | \( 1 + (-4.85 - 3.52i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (1.88 - 5.79i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-5.04 - 3.66i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (0.190 - 0.138i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 6.70T + 43T^{2} \) |
| 47 | \( 1 + (-8.16 + 5.93i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (0.118 - 0.363i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (5.97 + 4.33i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (3.57 + 10.9i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 - 1.85T + 67T^{2} \) |
| 71 | \( 1 + (-3.19 - 9.82i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-4.61 - 3.35i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-3.39 + 10.4i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-0.454 - 1.40i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + 8.23T + 89T^{2} \) |
| 97 | \( 1 + (-2.42 + 7.46i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−16.91882142420956781373832330125, −15.91332035097733474512034777870, −14.94034050981654425889176507722, −13.98210287115459103407761090817, −12.71538019834014812581411881352, −10.29314469740978559945928134412, −8.727398947388126533120623200704, −8.012237238379769935991718337682, −6.61462173149129510325974246137, −5.02455153538065297246012103544,
2.62604570344948288392588188459, 4.39487915900050490447519620477, 7.923647101006141623159654241072, 9.148644582998023905922649744957, 10.43942286907456188046405755049, 11.14766597732641612746030580536, 12.68301172593896206414019449094, 13.63358303594250610660446094091, 15.17156313286988885866690848012, 16.95132035947679451569949978950