L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s + (2.05 − 0.873i)5-s + (−0.499 + 0.866i)6-s − 2.32i·7-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (−1.34 + 1.78i)10-s − 2.85·11-s − 0.999i·12-s + (5.20 + 3.00i)13-s + (1.16 + 2.01i)14-s + (1.34 − 1.78i)15-s + (−0.5 − 0.866i)16-s + (3.79 − 2.19i)17-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.499 − 0.288i)3-s + (0.249 − 0.433i)4-s + (0.920 − 0.390i)5-s + (−0.204 + 0.353i)6-s − 0.877i·7-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.425 + 0.564i)10-s − 0.859·11-s − 0.288i·12-s + (1.44 + 0.833i)13-s + (0.310 + 0.537i)14-s + (0.347 − 0.461i)15-s + (−0.125 − 0.216i)16-s + (0.920 − 0.531i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.742 + 0.669i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.742 + 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.39403 - 0.535966i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.39403 - 0.535966i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 5 | \( 1 + (-2.05 + 0.873i)T \) |
| 19 | \( 1 + (2.63 + 3.47i)T \) |
good | 7 | \( 1 + 2.32iT - 7T^{2} \) |
| 11 | \( 1 + 2.85T + 11T^{2} \) |
| 13 | \( 1 + (-5.20 - 3.00i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.79 + 2.19i)T + (8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (4.36 + 2.51i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (4.41 - 7.63i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 0.685T + 31T^{2} \) |
| 37 | \( 1 + 7.79iT - 37T^{2} \) |
| 41 | \( 1 + (-2.58 - 4.47i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.24 + 1.29i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-6.83 - 3.94i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.86 + 2.81i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (0.724 + 1.25i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.40 + 9.35i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.89 - 3.40i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.15 - 1.99i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (5.38 - 3.10i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.00 - 3.47i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 12.2iT - 83T^{2} \) |
| 89 | \( 1 + (4.34 - 7.52i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.79 + 3.34i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.54111936086907313819061326670, −9.555917465500202133703367799771, −8.903184181751287866089086223919, −8.053941467248039539488701939383, −7.12438916073994765741600461602, −6.27818610332271147882782855339, −5.25674356025646729916916673932, −3.90711704883335121772727304016, −2.33194560803818436480698668085, −1.07846705924031717969005969352,
1.74742285424802915658289820323, 2.77786174369532756682261092203, 3.79965113326359950950662947601, 5.67239501522788238834664722808, 6.01166112530130951780292185737, 7.67953936681888708861538025075, 8.317557855480967627208389101303, 9.125556164972064848699322829201, 10.16597336670516927390057083812, 10.40241022657624409754774818253