L(s) = 1 | + (0.0342 + 0.216i)3-s + (−0.388 − 2.20i)5-s + (−0.104 − 0.0165i)7-s + (2.80 − 0.912i)9-s + (0.733 − 3.23i)11-s + (1.77 + 0.904i)13-s + (0.463 − 0.159i)15-s + (−4.77 + 2.43i)17-s + (−3.72 − 2.70i)19-s − 0.0232i·21-s + (1.26 + 1.26i)23-s + (−4.69 + 1.71i)25-s + (0.592 + 1.16i)27-s + (3.14 − 2.28i)29-s + (−2.80 − 8.62i)31-s + ⋯ |
L(s) = 1 | + (0.0197 + 0.124i)3-s + (−0.173 − 0.984i)5-s + (−0.0396 − 0.00627i)7-s + (0.935 − 0.304i)9-s + (0.221 − 0.975i)11-s + (0.492 + 0.250i)13-s + (0.119 − 0.0412i)15-s + (−1.15 + 0.590i)17-s + (−0.855 − 0.621i)19-s − 0.00507i·21-s + (0.263 + 0.263i)23-s + (−0.939 + 0.342i)25-s + (0.113 + 0.223i)27-s + (0.583 − 0.423i)29-s + (−0.503 − 1.54i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0484 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0484 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.02967 - 0.980910i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.02967 - 0.980910i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.388 + 2.20i)T \) |
| 11 | \( 1 + (-0.733 + 3.23i)T \) |
good | 3 | \( 1 + (-0.0342 - 0.216i)T + (-2.85 + 0.927i)T^{2} \) |
| 7 | \( 1 + (0.104 + 0.0165i)T + (6.65 + 2.16i)T^{2} \) |
| 13 | \( 1 + (-1.77 - 0.904i)T + (7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (4.77 - 2.43i)T + (9.99 - 13.7i)T^{2} \) |
| 19 | \( 1 + (3.72 + 2.70i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-1.26 - 1.26i)T + 23iT^{2} \) |
| 29 | \( 1 + (-3.14 + 2.28i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (2.80 + 8.62i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-1.42 + 9.02i)T + (-35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (-0.361 + 0.498i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (-2.10 + 2.10i)T - 43iT^{2} \) |
| 47 | \( 1 + (-8.76 + 1.38i)T + (44.6 - 14.5i)T^{2} \) |
| 53 | \( 1 + (-3.65 + 7.17i)T + (-31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (-6.35 - 8.75i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (0.692 + 0.225i)T + (49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (1.43 - 1.43i)T - 67iT^{2} \) |
| 71 | \( 1 + (0.219 - 0.674i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (2.08 - 13.1i)T + (-69.4 - 22.5i)T^{2} \) |
| 79 | \( 1 + (2.31 + 7.12i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-2.58 - 5.07i)T + (-48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 - 0.464iT - 89T^{2} \) |
| 97 | \( 1 + (1.58 + 0.809i)T + (57.0 + 78.4i)T^{2} \) |
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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
−9.804923355616576193656240288955, −8.911805004366803089756669978994, −8.565884999463652229443617585863, −7.40609103903363582840970728825, −6.42084086906073624506305999287, −5.57748970898845019253470714267, −4.26984727073587285504415114099, −3.92674176242326444755163366853, −2.12934756127681757883596082364, −0.70552379528220583721263248010,
1.68848917039983804363998559427, 2.83645805869843121939343867986, 4.07668284137183188554281942976, 4.86104847542611746966628628797, 6.37858942316404650015720957874, 6.85991695645786005885224776983, 7.63625333789410134219588188284, 8.630095641565150557315373181707, 9.647234292606696747223652621249, 10.48630716905484350475691936636