L(s) = 1 | + (0.707 − 1.22i)2-s + (−0.999 − 1.73i)4-s + (−4.43 − 2.30i)5-s + (6.62 + 3.82i)7-s − 2.82·8-s + (−5.96 + 3.80i)10-s + (9.85 + 5.68i)11-s + (−12.4 + 7.19i)13-s + (9.36 − 5.40i)14-s + (−2.00 + 3.46i)16-s + 17.3·17-s − 31.8·19-s + (0.445 + 9.99i)20-s + (13.9 − 8.04i)22-s + (8.25 + 14.3i)23-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.887 − 0.460i)5-s + (0.946 + 0.546i)7-s − 0.353·8-s + (−0.596 + 0.380i)10-s + (0.895 + 0.517i)11-s + (−0.958 + 0.553i)13-s + (0.668 − 0.386i)14-s + (−0.125 + 0.216i)16-s + 1.01·17-s − 1.67·19-s + (0.0222 + 0.499i)20-s + (0.633 − 0.365i)22-s + (0.359 + 0.621i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.131i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.991 - 0.131i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.821142051\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.821142051\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 1.22i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (4.43 + 2.30i)T \) |
good | 7 | \( 1 + (-6.62 - 3.82i)T + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-9.85 - 5.68i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (12.4 - 7.19i)T + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 - 17.3T + 289T^{2} \) |
| 19 | \( 1 + 31.8T + 361T^{2} \) |
| 23 | \( 1 + (-8.25 - 14.3i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (7.35 + 4.24i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-22.9 - 39.6i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 31.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-59.2 + 34.1i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-27.1 - 15.6i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (23.0 - 39.8i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 - 53.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-56.0 + 32.3i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-48.1 + 83.3i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-1.49 + 0.862i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 26.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 51.4iT - 5.32e3T^{2} \) |
| 79 | \( 1 + (29.6 - 51.2i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-3.46 + 6.00i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 115. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-137. - 79.3i)T + (4.70e3 + 8.14e3i)T^{2} \) |
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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
−10.13361522189245154350339350752, −9.200675647421726602825441405200, −8.496283294411113004926818519451, −7.60878452480896469607840121955, −6.58293004357697740123540400737, −5.24573907364794356622721890299, −4.58172470107390752616829721842, −3.78069441850720208423630235735, −2.33892246131872819060929060526, −1.20992926498332012032806996829,
0.63306500496743919677026638919, 2.59260198869834425814068619527, 3.94388300588664923066494950113, 4.43344601468316923271901802525, 5.65290437836351916759623636863, 6.66147203675095310438442316668, 7.51999494617659944900614023165, 8.058449983275806303913114713651, 8.883925064385939401724497937548, 10.17188173243432288435545272979