| L(s) = 1 | + (−0.249 − 1.71i)3-s + (−1.47 + 1.75i)5-s + (−0.590 + 1.62i)7-s + (−2.87 + 0.854i)9-s + (0.867 − 0.728i)11-s + (−1.22 + 6.94i)13-s + (3.38 + 2.09i)15-s + (−1.48 − 0.857i)17-s + (−3.94 + 2.27i)19-s + (2.92 + 0.608i)21-s + (6.51 − 2.37i)23-s + (−0.0475 − 0.269i)25-s + (2.18 + 4.71i)27-s + (−6.35 + 1.12i)29-s + (1.56 + 4.30i)31-s + ⋯ |
| L(s) = 1 | + (−0.143 − 0.989i)3-s + (−0.660 + 0.786i)5-s + (−0.223 + 0.613i)7-s + (−0.958 + 0.284i)9-s + (0.261 − 0.219i)11-s + (−0.339 + 1.92i)13-s + (0.873 + 0.540i)15-s + (−0.360 − 0.207i)17-s + (−0.904 + 0.522i)19-s + (0.639 + 0.132i)21-s + (1.35 − 0.494i)23-s + (−0.00950 − 0.0538i)25-s + (0.419 + 0.907i)27-s + (−1.17 + 0.208i)29-s + (0.281 + 0.773i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0323 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0323 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.522666 + 0.506009i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.522666 + 0.506009i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.249 + 1.71i)T \) |
| good | 5 | \( 1 + (1.47 - 1.75i)T + (-0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (0.590 - 1.62i)T + (-5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (-0.867 + 0.728i)T + (1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (1.22 - 6.94i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (1.48 + 0.857i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.94 - 2.27i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-6.51 + 2.37i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (6.35 - 1.12i)T + (27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-1.56 - 4.30i)T + (-23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (-1.47 + 2.55i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-7.14 - 1.25i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (0.337 + 0.402i)T + (-7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (8.97 + 3.26i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 - 8.57iT - 53T^{2} \) |
| 59 | \( 1 + (11.0 + 9.25i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (0.269 + 0.0979i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (5.85 + 1.03i)T + (62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-1.36 + 2.36i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (1.58 + 2.74i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (9.41 - 1.65i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-0.731 - 4.14i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (5.83 - 3.36i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (3.21 - 2.69i)T + (16.8 - 95.5i)T^{2} \) |
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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
−11.32806057225992626896815040609, −10.95474343271966120765652534317, −9.300722536901324554921208095146, −8.671005335266558037367438077761, −7.42041284811471256525028488627, −6.80803338695062439120111499575, −6.04707241716501969417092734430, −4.54223915001020200027989669081, −3.12309675785696281445747454952, −1.91969191828435945766313973309,
0.45992030437201958810298602596, 3.03203782895687957010529776923, 4.14623093717478039479388664994, 4.91126009504041927081351801061, 5.96998433240520484452466798887, 7.40270445967156399738271144988, 8.310835978359908701929164248087, 9.168396118022320800318120572002, 10.09667778808112050813393471863, 10.86694332064711209451683596062
