| L(s) = 1 | + (0.707 + 1.22i)2-s + (−0.999 + 1.73i)4-s + (−7.24 + 4.18i)5-s + (−6.74 − 1.88i)7-s − 2.82·8-s + (−10.2 − 5.91i)10-s + (3 − 5.19i)11-s + 17.8i·13-s + (−2.46 − 9.58i)14-s + (−2.00 − 3.46i)16-s + (16.2 + 9.37i)17-s + (−14.7 + 8.51i)19-s − 16.7i·20-s + 8.48·22-s + (6.72 + 11.6i)23-s + ⋯ |
| L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−1.44 + 0.836i)5-s + (−0.963 − 0.268i)7-s − 0.353·8-s + (−1.02 − 0.591i)10-s + (0.272 − 0.472i)11-s + 1.37i·13-s + (−0.176 − 0.684i)14-s + (−0.125 − 0.216i)16-s + (0.955 + 0.551i)17-s + (−0.775 + 0.447i)19-s − 0.836i·20-s + 0.385·22-s + (0.292 + 0.506i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.144i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.989 - 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0537770 + 0.739153i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0537770 + 0.739153i\) |
| \(L(2)\) |
|
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 | \( 1 \) |
| 7 | \( 1 + (6.74 + 1.88i)T \) |
| good | 5 | \( 1 + (7.24 - 4.18i)T + (12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (-3 + 5.19i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 - 17.8iT - 169T^{2} \) |
| 17 | \( 1 + (-16.2 - 9.37i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (14.7 - 8.51i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-6.72 - 11.6i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + 33.9T + 841T^{2} \) |
| 31 | \( 1 + (-12.7 - 7.37i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (2.98 + 5.17i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 - 35.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 15.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-28.7 + 16.6i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (17.2 - 29.9i)T + (-1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (23.6 + 13.6i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (34.9 - 20.1i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-57.1 + 99.0i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 18.6T + 5.04e3T^{2} \) |
| 73 | \( 1 + (101. + 58.5i)T + (2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-44.1 - 76.5i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 - 75.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-18 + 10.3i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + 30.5iT - 9.40e3T^{2} \) |
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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
−13.83634994338412684019641177639, −12.56539545939959439158337831855, −11.71069470592217364712138203689, −10.69849217526231397657747095311, −9.268959572161725507693014105160, −7.967205827201395179097672270862, −7.03938743408257599033833084683, −6.15613721402608089145109930584, −4.13748045058694122429008457519, −3.37087450268952251697580910419,
0.45728706469460177563151992409, 3.10300510715676986415934547553, 4.26517285066874729248914854187, 5.56361385139454075428257393573, 7.27856737918960901944340489852, 8.462407142832589518286540870139, 9.544254091697781155379272964989, 10.73672234666713314625604798567, 11.93053035131645276652041567051, 12.54541276688741425945192202661
