| L(s) = 1 | + (−0.5 + 0.133i)2-s + (0.366 − 0.633i)3-s + (−3.23 + 1.86i)4-s + (−2.63 − 2.63i)5-s + (−0.0980 + 0.366i)6-s + (5.73 + 1.53i)7-s + (2.83 − 2.83i)8-s + (4.23 + 7.33i)9-s + (1.66 + 0.964i)10-s + (−4.19 − 15.6i)11-s + 2.73i·12-s + (−6.5 + 11.2i)13-s − 3.07·14-s + (−2.63 + 0.705i)15-s + (6.42 − 11.1i)16-s + (−15.9 + 9.23i)17-s + ⋯ |
| L(s) = 1 | + (−0.250 + 0.0669i)2-s + (0.122 − 0.211i)3-s + (−0.808 + 0.466i)4-s + (−0.526 − 0.526i)5-s + (−0.0163 + 0.0610i)6-s + (0.818 + 0.219i)7-s + (0.353 − 0.353i)8-s + (0.470 + 0.814i)9-s + (0.166 + 0.0964i)10-s + (−0.381 − 1.42i)11-s + 0.227i·12-s + (−0.5 + 0.866i)13-s − 0.219·14-s + (−0.175 + 0.0470i)15-s + (0.401 − 0.695i)16-s + (−0.940 + 0.543i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0257i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0257i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.619000 + 0.00798248i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.619000 + 0.00798248i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (6.5 - 11.2i)T \) |
| good | 2 | \( 1 + (0.5 - 0.133i)T + (3.46 - 2i)T^{2} \) |
| 3 | \( 1 + (-0.366 + 0.633i)T + (-4.5 - 7.79i)T^{2} \) |
| 5 | \( 1 + (2.63 + 2.63i)T + 25iT^{2} \) |
| 7 | \( 1 + (-5.73 - 1.53i)T + (42.4 + 24.5i)T^{2} \) |
| 11 | \( 1 + (4.19 + 15.6i)T + (-104. + 60.5i)T^{2} \) |
| 17 | \( 1 + (15.9 - 9.23i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-1.63 + 6.09i)T + (-312. - 180.5i)T^{2} \) |
| 23 | \( 1 + (-17.4 - 10.0i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (4.69 - 8.13i)T + (-420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (11.9 + 11.9i)T + 961iT^{2} \) |
| 37 | \( 1 + (-8.11 - 30.2i)T + (-1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + (-44.9 + 12.0i)T + (1.45e3 - 840.5i)T^{2} \) |
| 43 | \( 1 + (-45 + 25.9i)T + (924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (34.3 - 34.3i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + 14.7T + 2.80e3T^{2} \) |
| 59 | \( 1 + (92.9 + 24.9i)T + (3.01e3 + 1.74e3i)T^{2} \) |
| 61 | \( 1 + (12.8 + 22.1i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-39.0 + 10.4i)T + (3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 + (11.9 - 44.6i)T + (-4.36e3 - 2.52e3i)T^{2} \) |
| 73 | \( 1 + (19.2 - 19.2i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 - 62.7T + 6.24e3T^{2} \) |
| 83 | \( 1 + (24.4 + 24.4i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + (23.1 + 86.4i)T + (-6.85e3 + 3.96e3i)T^{2} \) |
| 97 | \( 1 + (-14.1 + 52.9i)T + (-8.14e3 - 4.70e3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−19.46363117579604263318838230277, −18.56541507849067410742095645973, −17.13365449821973997528271361000, −15.99527459313062271554975282177, −14.02871999898276824260912970034, −12.87656638065065882475295154215, −11.17170965216970607950473066331, −8.882317382453934682412255272055, −7.82745848784865698386406467506, −4.65942459653037270782916514650,
4.61993023668330184227022435093, 7.51474711131236354201642582585, 9.441057435701829335531531376602, 10.80114878676248533836218564486, 12.73506263469685486997756218993, 14.60793667814246444314596565503, 15.29071013712631202961537634294, 17.64350259213821990500938318148, 18.21182317074686779649978089077, 19.76991606705878862998868980616
