| L(s) = 1 | + (−0.587 − 1.28i)2-s + (−4.52 + 1.33i)3-s + (−1.30 + 1.51i)4-s + (−1.16 + 1.81i)5-s + (4.37 + 5.04i)6-s + (−10.6 + 1.53i)7-s + (2.71 + 0.796i)8-s + (11.1 − 7.18i)9-s + (3.02 + 0.434i)10-s + (1.02 + 0.468i)11-s + (3.92 − 8.58i)12-s + (2.02 − 14.1i)13-s + (8.22 + 12.8i)14-s + (2.87 − 9.78i)15-s + (−0.569 − 3.95i)16-s + (−21.2 + 18.3i)17-s + ⋯ |
| L(s) = 1 | + (−0.293 − 0.643i)2-s + (−1.50 + 0.443i)3-s + (−0.327 + 0.377i)4-s + (−0.233 + 0.363i)5-s + (0.728 + 0.840i)6-s + (−1.52 + 0.218i)7-s + (0.339 + 0.0996i)8-s + (1.24 − 0.798i)9-s + (0.302 + 0.0434i)10-s + (0.0932 + 0.0425i)11-s + (0.326 − 0.715i)12-s + (0.155 − 1.08i)13-s + (0.587 + 0.914i)14-s + (0.191 − 0.652i)15-s + (−0.0355 − 0.247i)16-s + (−1.24 + 1.08i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.774 - 0.632i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.774 - 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0425403 + 0.119329i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0425403 + 0.119329i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.587 + 1.28i)T \) |
| 23 | \( 1 + (-3.63 - 22.7i)T \) |
| good | 3 | \( 1 + (4.52 - 1.33i)T + (7.57 - 4.86i)T^{2} \) |
| 5 | \( 1 + (1.16 - 1.81i)T + (-10.3 - 22.7i)T^{2} \) |
| 7 | \( 1 + (10.6 - 1.53i)T + (47.0 - 13.8i)T^{2} \) |
| 11 | \( 1 + (-1.02 - 0.468i)T + (79.2 + 91.4i)T^{2} \) |
| 13 | \( 1 + (-2.02 + 14.1i)T + (-162. - 47.6i)T^{2} \) |
| 17 | \( 1 + (21.2 - 18.3i)T + (41.1 - 286. i)T^{2} \) |
| 19 | \( 1 + (2.49 + 2.16i)T + (51.3 + 357. i)T^{2} \) |
| 29 | \( 1 + (-25.8 - 29.8i)T + (-119. + 832. i)T^{2} \) |
| 31 | \( 1 + (47.2 + 13.8i)T + (808. + 519. i)T^{2} \) |
| 37 | \( 1 + (34.2 + 53.2i)T + (-568. + 1.24e3i)T^{2} \) |
| 41 | \( 1 + (36.7 + 23.6i)T + (698. + 1.52e3i)T^{2} \) |
| 43 | \( 1 + (-4.62 - 15.7i)T + (-1.55e3 + 9.99e2i)T^{2} \) |
| 47 | \( 1 - 25.6T + 2.20e3T^{2} \) |
| 53 | \( 1 + (26.9 - 3.87i)T + (2.69e3 - 791. i)T^{2} \) |
| 59 | \( 1 + (-0.462 + 3.21i)T + (-3.33e3 - 980. i)T^{2} \) |
| 61 | \( 1 + (7.17 - 24.4i)T + (-3.13e3 - 2.01e3i)T^{2} \) |
| 67 | \( 1 + (-8.81 + 4.02i)T + (2.93e3 - 3.39e3i)T^{2} \) |
| 71 | \( 1 + (-1.52 - 3.34i)T + (-3.30e3 + 3.80e3i)T^{2} \) |
| 73 | \( 1 + (25.2 - 29.1i)T + (-758. - 5.27e3i)T^{2} \) |
| 79 | \( 1 + (-23.8 - 3.43i)T + (5.98e3 + 1.75e3i)T^{2} \) |
| 83 | \( 1 + (-5.25 - 8.17i)T + (-2.86e3 + 6.26e3i)T^{2} \) |
| 89 | \( 1 + (-21.5 - 73.4i)T + (-6.66e3 + 4.28e3i)T^{2} \) |
| 97 | \( 1 + (67.3 - 104. i)T + (-3.90e3 - 8.55e3i)T^{2} \) |
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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
−16.11333692733018417830523545555, −15.32348219344667355554968240202, −13.06693843645966884350544469566, −12.39848550303588880569336960178, −11.01011496220594646640120463363, −10.44526935470166361950875192902, −9.135321586474494034639197924317, −6.88190034789161751802215818145, −5.58618698193511397720861842732, −3.60242851711535271545962559693,
0.16898266911048116385231458294, 4.68665466147351373175284090712, 6.40881819288544772031991270027, 6.85101723908890240965059314629, 8.956586110153737813811765357199, 10.30759154955356241633142366604, 11.65543261090613029005475456752, 12.69636815451027039387314563173, 13.76247720245703593153225411410, 15.75973717528270225388646598798
