L(s) = 1 | + (2.05 + 1.18i)5-s + (−4.05 − 7.02i)7-s + (−17.6 + 10.1i)11-s + (−3.05 + 5.29i)13-s + 17.9i·17-s − 9.11·19-s + (29.0 + 16.7i)23-s + (−9.67 − 16.7i)25-s + (−14.4 + 8.31i)29-s + (−11.1 + 19.3i)31-s − 19.2i·35-s − 50.4·37-s + (−29.9 − 17.3i)41-s + (11.5 + 19.9i)43-s + (−33.1 + 19.1i)47-s + ⋯ |
L(s) = 1 | + (0.411 + 0.237i)5-s + (−0.579 − 1.00i)7-s + (−1.60 + 0.924i)11-s + (−0.235 + 0.407i)13-s + 1.05i·17-s − 0.479·19-s + (1.26 + 0.729i)23-s + (−0.387 − 0.670i)25-s + (−0.496 + 0.286i)29-s + (−0.360 + 0.624i)31-s − 0.551i·35-s − 1.36·37-s + (−0.730 − 0.421i)41-s + (0.267 + 0.463i)43-s + (−0.705 + 0.407i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.800 - 0.598i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.800 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.164033 + 0.493409i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.164033 + 0.493409i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-2.05 - 1.18i)T + (12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (4.05 + 7.02i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (17.6 - 10.1i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (3.05 - 5.29i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 - 17.9iT - 289T^{2} \) |
| 19 | \( 1 + 9.11T + 361T^{2} \) |
| 23 | \( 1 + (-29.0 - 16.7i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (14.4 - 8.31i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (11.1 - 19.3i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 50.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + (29.9 + 17.3i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-11.5 - 19.9i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (33.1 - 19.1i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 - 19.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (2.96 + 1.71i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-23.1 - 40.1i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (3.14 - 5.45i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 35.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 47.3T + 5.32e3T^{2} \) |
| 79 | \( 1 + (42.2 + 73.2i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (33.1 - 19.1i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 143. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (40.3 + 69.9i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−10.97590133581524301361476328776, −10.34696537304586190932083472877, −9.792648264645538219393866574827, −8.557647380330883618915903576274, −7.40751368444858164295824442027, −6.83578039756376643305952054491, −5.57447918962639702510055295503, −4.49869597471401759577878814414, −3.24280240460918905607988370041, −1.87319335159163778490913242882,
0.20139520616062212223795453517, 2.36653335536625783726019126306, 3.22196477421003218479515465631, 5.15614356859366847743402684494, 5.54443163501940799332103508261, 6.76275454506900782842984875629, 7.954462800412299223774631351442, 8.828587322506198387686963673091, 9.604423012717981049494242312171, 10.57943134492028442830487929702