| L(s) = 1 | + (−0.126 + 8.99i)3-s + (−6.31 − 17.3i)5-s + (2.68 + 15.2i)7-s + (−80.9 − 2.27i)9-s + (−57.3 + 157. i)11-s + (−47.4 − 39.8i)13-s + (156. − 54.6i)15-s + (−434. − 250. i)17-s + (−226. − 391. i)19-s + (−137. + 22.2i)21-s + (−87.9 − 15.5i)23-s + (217. − 182. i)25-s + (30.6 − 728. i)27-s + (−388. − 462. i)29-s + (−253. + 1.43e3i)31-s + ⋯ |
| L(s) = 1 | + (−0.0140 + 0.999i)3-s + (−0.252 − 0.693i)5-s + (0.0548 + 0.310i)7-s + (−0.999 − 0.0280i)9-s + (−0.474 + 1.30i)11-s + (−0.280 − 0.235i)13-s + (0.697 − 0.242i)15-s + (−1.50 − 0.867i)17-s + (−0.626 − 1.08i)19-s + (−0.311 + 0.0504i)21-s + (−0.166 − 0.0293i)23-s + (0.348 − 0.292i)25-s + (0.0421 − 0.999i)27-s + (−0.461 − 0.550i)29-s + (−0.263 + 1.49i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 + 0.357i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.934 + 0.357i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.0406211 - 0.220028i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0406211 - 0.220028i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.126 - 8.99i)T \) |
| good | 5 | \( 1 + (6.31 + 17.3i)T + (-478. + 401. i)T^{2} \) |
| 7 | \( 1 + (-2.68 - 15.2i)T + (-2.25e3 + 821. i)T^{2} \) |
| 11 | \( 1 + (57.3 - 157. i)T + (-1.12e4 - 9.41e3i)T^{2} \) |
| 13 | \( 1 + (47.4 + 39.8i)T + (4.95e3 + 2.81e4i)T^{2} \) |
| 17 | \( 1 + (434. + 250. i)T + (4.17e4 + 7.23e4i)T^{2} \) |
| 19 | \( 1 + (226. + 391. i)T + (-6.51e4 + 1.12e5i)T^{2} \) |
| 23 | \( 1 + (87.9 + 15.5i)T + (2.62e5 + 9.57e4i)T^{2} \) |
| 29 | \( 1 + (388. + 462. i)T + (-1.22e5 + 6.96e5i)T^{2} \) |
| 31 | \( 1 + (253. - 1.43e3i)T + (-8.67e5 - 3.15e5i)T^{2} \) |
| 37 | \( 1 + (743. - 1.28e3i)T + (-9.37e5 - 1.62e6i)T^{2} \) |
| 41 | \( 1 + (1.17e3 - 1.40e3i)T + (-4.90e5 - 2.78e6i)T^{2} \) |
| 43 | \( 1 + (-1.14e3 - 416. i)T + (2.61e6 + 2.19e6i)T^{2} \) |
| 47 | \( 1 + (1.67e3 - 295. i)T + (4.58e6 - 1.66e6i)T^{2} \) |
| 53 | \( 1 + 448. iT - 7.89e6T^{2} \) |
| 59 | \( 1 + (-519. - 1.42e3i)T + (-9.28e6 + 7.78e6i)T^{2} \) |
| 61 | \( 1 + (-600. - 3.40e3i)T + (-1.30e7 + 4.73e6i)T^{2} \) |
| 67 | \( 1 + (-130. - 109. i)T + (3.49e6 + 1.98e7i)T^{2} \) |
| 71 | \( 1 + (3.37e3 + 1.95e3i)T + (1.27e7 + 2.20e7i)T^{2} \) |
| 73 | \( 1 + (-4.20e3 - 7.28e3i)T + (-1.41e7 + 2.45e7i)T^{2} \) |
| 79 | \( 1 + (-6.49e3 + 5.44e3i)T + (6.76e6 - 3.83e7i)T^{2} \) |
| 83 | \( 1 + (697. + 831. i)T + (-8.24e6 + 4.67e7i)T^{2} \) |
| 89 | \( 1 + (9.94e3 - 5.74e3i)T + (3.13e7 - 5.43e7i)T^{2} \) |
| 97 | \( 1 + (1.07e3 + 392. i)T + (6.78e7 + 5.69e7i)T^{2} \) |
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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
−13.51233640986509529967918294965, −12.45834707183115983319495630126, −11.42572909043823275253269945019, −10.31885457118880993297797013478, −9.261407086402254544153739769046, −8.451770039301093983470583985982, −6.87508362854958523959908385174, −5.04492817407753677538164032643, −4.48973471043978698784955942420, −2.54769144271721128208891505111,
0.092483937099869574057611231077, 2.11319985477552428529796924779, 3.68115393350350115554225045153, 5.77546580892794031928104996315, 6.77409680468413433366803602980, 7.86991043380046167767605643436, 8.837498175329183714412859368120, 10.70881575478745441807667068937, 11.20864459564078116550295930759, 12.53564199198411466829948104826
