L(s) = 1 | + (0.262 − 0.454i)2-s + (0.866 − 0.5i)3-s + (0.862 + 1.49i)4-s + (−3.41 − 1.97i)5-s − 0.524i·6-s + (2.63 − 0.177i)7-s + 1.95·8-s + (0.499 − 0.866i)9-s + (−1.79 + 1.03i)10-s + (2.72 − 1.57i)11-s + (1.49 + 0.862i)12-s + 0.881·13-s + (0.611 − 1.24i)14-s − 3.94·15-s + (−1.21 + 2.09i)16-s + (1.84 − 3.68i)17-s + ⋯ |
L(s) = 1 | + (0.185 − 0.321i)2-s + (0.499 − 0.288i)3-s + (0.431 + 0.746i)4-s + (−1.52 − 0.882i)5-s − 0.214i·6-s + (0.997 − 0.0671i)7-s + 0.691·8-s + (0.166 − 0.288i)9-s + (−0.567 + 0.327i)10-s + (0.821 − 0.474i)11-s + (0.431 + 0.248i)12-s + 0.244·13-s + (0.163 − 0.333i)14-s − 1.01·15-s + (−0.302 + 0.524i)16-s + (0.446 − 0.894i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.655 + 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.655 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.58324 - 0.722374i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.58324 - 0.722374i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 7 | \( 1 + (-2.63 + 0.177i)T \) |
| 17 | \( 1 + (-1.84 + 3.68i)T \) |
good | 2 | \( 1 + (-0.262 + 0.454i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (3.41 + 1.97i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.72 + 1.57i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 0.881T + 13T^{2} \) |
| 19 | \( 1 + (-1.76 + 3.06i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.87 + 1.65i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 8.73iT - 29T^{2} \) |
| 31 | \( 1 + (3.05 - 1.76i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (10.3 + 5.98i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 6.52iT - 41T^{2} \) |
| 43 | \( 1 - 3.56T + 43T^{2} \) |
| 47 | \( 1 + (4.34 - 7.53i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.38 - 4.13i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (6.39 + 11.0i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.80 - 3.92i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.65 + 2.86i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 4.10iT - 71T^{2} \) |
| 73 | \( 1 + (9.52 - 5.50i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.03 - 1.75i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 + (-0.556 + 0.964i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 4.98iT - 97T^{2} \) |
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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
−11.51003780527628155021779919612, −10.93298463540694205015564011289, −9.023699786590689240525482207235, −8.470623173128621758527164273794, −7.64466203904393557977710694208, −7.02648230698524251511873424166, −5.01603397638725231463547044876, −4.05136471545100269061000800570, −3.18320262830225977610534022076, −1.33320166115611011257557441541,
1.83957328741200693843299304103, 3.62227609161636817200007119401, 4.39105369061159656652981458181, 5.79116522903046160132448761868, 7.02370080689918151665436582900, 7.71638101953854417895571573475, 8.496897790982751649126543465504, 9.999199500881568120275617436161, 10.66511660919621862005236586458, 11.63048154706598337402200069172