L(s) = 1 | + (−0.5 − 0.866i)2-s + (1.72 − 0.162i)3-s + (−0.499 + 0.866i)4-s + (0.866 − 0.5i)5-s + (−1.00 − 1.41i)6-s + 3.36·7-s + 0.999·8-s + (2.94 − 0.561i)9-s + (−0.866 − 0.499i)10-s + 0.795i·11-s + (−0.721 + 1.57i)12-s + (1.59 + 0.922i)13-s + (−1.68 − 2.91i)14-s + (1.41 − 1.00i)15-s + (−0.5 − 0.866i)16-s + (−6.17 + 3.56i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.995 − 0.0939i)3-s + (−0.249 + 0.433i)4-s + (0.387 − 0.223i)5-s + (−0.409 − 0.576i)6-s + 1.27·7-s + 0.353·8-s + (0.982 − 0.187i)9-s + (−0.273 − 0.158i)10-s + 0.239i·11-s + (−0.208 + 0.454i)12-s + (0.443 + 0.255i)13-s + (−0.449 − 0.779i)14-s + (0.364 − 0.259i)15-s + (−0.125 − 0.216i)16-s + (−1.49 + 0.865i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.752 + 0.658i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.752 + 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.87161 - 0.703617i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.87161 - 0.703617i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 + (-1.72 + 0.162i)T \) |
| 5 | \( 1 + (-0.866 + 0.5i)T \) |
| 19 | \( 1 + (0.831 - 4.27i)T \) |
good | 7 | \( 1 - 3.36T + 7T^{2} \) |
| 11 | \( 1 - 0.795iT - 11T^{2} \) |
| 13 | \( 1 + (-1.59 - 0.922i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (6.17 - 3.56i)T + (8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (1.72 + 0.997i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.95 + 3.37i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 8.28iT - 31T^{2} \) |
| 37 | \( 1 - 3.63iT - 37T^{2} \) |
| 41 | \( 1 + (2.08 + 3.61i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.75 + 3.03i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (3.13 + 1.80i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.21 + 5.56i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.40 - 7.62i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.34 - 5.80i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.16 + 1.82i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.800 - 1.38i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (7.05 + 12.2i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (10.3 - 5.97i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 13.9iT - 83T^{2} \) |
| 89 | \( 1 + (1.83 - 3.18i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (15.9 - 9.19i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.49868714340263291900600089510, −9.788023820595025926768552997333, −8.615455913254154124141377246366, −8.430781705911927868198038382838, −7.41288488589663101463087949991, −6.14325067033159692636221543608, −4.58613316196062775156976789821, −3.92054806319785561275818650516, −2.26606247165588683611625820309, −1.62251441672412676404855265629,
1.57243614971032717757109471937, 2.83074386570320039646389391408, 4.41119908739430887010625138167, 5.16666871274970020871814811313, 6.62349969169094531142969029117, 7.32755938385154938034405541086, 8.423698613179889148682671613953, 8.768583039123595938515145970375, 9.717927924702171499360374475326, 10.77640727770169291530274428981