L(s) = 1 | + (0.707 − 0.707i)2-s + (−1.31 + 1.12i)3-s − 1.00i·4-s + (−0.258 − 0.965i)5-s + (−0.136 + 1.72i)6-s + (−0.443 + 2.60i)7-s + (−0.707 − 0.707i)8-s + (0.470 − 2.96i)9-s + (−0.865 − 0.499i)10-s + (2.62 − 0.704i)11-s + (1.12 + 1.31i)12-s + (3.59 + 0.238i)13-s + (1.53 + 2.15i)14-s + (1.42 + 0.981i)15-s − 1.00·16-s + 2.39·17-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (−0.760 + 0.649i)3-s − 0.500i·4-s + (−0.115 − 0.431i)5-s + (−0.0556 + 0.704i)6-s + (−0.167 + 0.985i)7-s + (−0.250 − 0.250i)8-s + (0.156 − 0.987i)9-s + (−0.273 − 0.158i)10-s + (0.792 − 0.212i)11-s + (0.324 + 0.380i)12-s + (0.997 + 0.0661i)13-s + (0.409 + 0.576i)14-s + (0.368 + 0.253i)15-s − 0.250·16-s + 0.580·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.209i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 + 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.53258 - 0.162628i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.53258 - 0.162628i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 + (1.31 - 1.12i)T \) |
| 7 | \( 1 + (0.443 - 2.60i)T \) |
| 13 | \( 1 + (-3.59 - 0.238i)T \) |
good | 5 | \( 1 + (0.258 + 0.965i)T + (-4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (-2.62 + 0.704i)T + (9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 - 2.39T + 17T^{2} \) |
| 19 | \( 1 + (0.807 - 3.01i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 - 6.02T + 23T^{2} \) |
| 29 | \( 1 + (-2.49 + 1.43i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (0.354 - 1.32i)T + (-26.8 - 15.5i)T^{2} \) |
| 37 | \( 1 + (-0.770 + 0.770i)T - 37iT^{2} \) |
| 41 | \( 1 + (0.406 + 0.108i)T + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (3.05 + 1.76i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.717 + 0.192i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (2.12 - 1.22i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-7.69 - 7.69i)T + 59iT^{2} \) |
| 61 | \( 1 + (0.903 + 1.56i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.116 - 0.0311i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (3.68 + 13.7i)T + (-61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (-4.19 - 1.12i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (5.71 - 9.90i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (7.50 - 7.50i)T - 83iT^{2} \) |
| 89 | \( 1 + (7.22 + 7.22i)T + 89iT^{2} \) |
| 97 | \( 1 + (4.73 - 1.26i)T + (84.0 - 48.5i)T^{2} \) |
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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
−10.95515601943021057152329533948, −10.04254742169690323410291244554, −9.109274573728827934750940142599, −8.518179504159250987128376274664, −6.69848357193673925502836543581, −5.91987641560020379524668465897, −5.14325439131355427827006718107, −4.09954996106298038004896461572, −3.11812918366163257804806090225, −1.20765146743833754914619842389,
1.15698919045244015206561708210, 3.14676926020871760523626231772, 4.30489948826815058111260679864, 5.35108565638502260562236576879, 6.58603469121550726114407344082, 6.85047086992523896965555478906, 7.78091528303281187978395671761, 8.877980658958725355776197760400, 10.20756596884361812418357826154, 11.09431780288134724827516530351