| L(s) = 1 | + (−1.70 − 0.983i)5-s + (−8.69 + 5.02i)7-s + (6.08 + 10.5i)11-s + (−4.28 − 2.47i)13-s − 4.71·17-s + 20.5·19-s + (3.33 + 1.92i)23-s + (−10.5 − 18.2i)25-s + (40.7 − 23.5i)29-s + (−49.9 − 28.8i)31-s + 19.7·35-s − 7.93i·37-s + (11.3 − 19.6i)41-s + (−30.7 − 53.3i)43-s + (−44.7 + 25.8i)47-s + ⋯ |
| L(s) = 1 | + (−0.340 − 0.196i)5-s + (−1.24 + 0.717i)7-s + (0.553 + 0.958i)11-s + (−0.329 − 0.190i)13-s − 0.277·17-s + 1.08·19-s + (0.144 + 0.0836i)23-s + (−0.422 − 0.731i)25-s + (1.40 − 0.811i)29-s + (−1.61 − 0.929i)31-s + 0.564·35-s − 0.214i·37-s + (0.276 − 0.479i)41-s + (−0.715 − 1.23i)43-s + (−0.951 + 0.549i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.210 + 0.977i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.210 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7097022023\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7097022023\) |
| \(L(2)\) |
|
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 | \( 1 + (1.70 + 0.983i)T + (12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (8.69 - 5.02i)T + (24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-6.08 - 10.5i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (4.28 + 2.47i)T + (84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + 4.71T + 289T^{2} \) |
| 19 | \( 1 - 20.5T + 361T^{2} \) |
| 23 | \( 1 + (-3.33 - 1.92i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-40.7 + 23.5i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (49.9 + 28.8i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 7.93iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-11.3 + 19.6i)T + (-840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (30.7 + 53.3i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (44.7 - 25.8i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + 51.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-16.7 + 28.9i)T + (-1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (39.7 - 22.9i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (26.9 - 46.6i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 132. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 24.6T + 5.32e3T^{2} \) |
| 79 | \( 1 + (84.3 - 48.6i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-0.187 - 0.324i)T + (-3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 - 134.T + 7.92e3T^{2} \) |
| 97 | \( 1 + (-10.8 - 18.8i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−9.603598799160791432649027781879, −9.125418342627681454048180974390, −8.013508421234850339446539762153, −7.09759059726365655501627962119, −6.31107227709037304925882226934, −5.34999620689199762778574369656, −4.24599578095827898712308586369, −3.22542032949661982406273550871, −2.09234942907346794236391615282, −0.25898533794451765491728092080,
1.16181465555763555020568233735, 3.12070193409928635179779912938, 3.54665544425012061284387511785, 4.82649784978939725791833884332, 6.02588420980357337612833123131, 6.83289599152195675929390150807, 7.45739203000009736233436281909, 8.632966295934537620711440470494, 9.413282079799847730902425787081, 10.15217348069117967085121395211
