| L(s) = 1 | + (−0.938 + 0.627i)3-s + (−4.64 − 0.924i)5-s + (0.473 − 0.0942i)7-s + (−2.95 + 7.13i)9-s + (−11.0 + 16.5i)11-s + (−13.5 − 13.5i)13-s + (4.94 − 2.04i)15-s + (−13.7 − 10.0i)17-s + (1.11 + 2.69i)19-s + (−0.385 + 0.385i)21-s + (31.9 + 21.3i)23-s + (−2.36 − 0.980i)25-s + (−3.68 − 18.5i)27-s + (−4.99 + 25.1i)29-s + (−3.12 − 4.67i)31-s + ⋯ |
| L(s) = 1 | + (−0.312 + 0.209i)3-s + (−0.929 − 0.184i)5-s + (0.0676 − 0.0134i)7-s + (−0.328 + 0.793i)9-s + (−1.00 + 1.50i)11-s + (−1.04 − 1.04i)13-s + (0.329 − 0.136i)15-s + (−0.806 − 0.590i)17-s + (0.0586 + 0.141i)19-s + (−0.0183 + 0.0183i)21-s + (1.38 + 0.927i)23-s + (−0.0946 − 0.0392i)25-s + (−0.136 − 0.685i)27-s + (−0.172 + 0.866i)29-s + (−0.100 − 0.150i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.924 - 0.381i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.924 - 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0643285 + 0.324504i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0643285 + 0.324504i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + (13.7 + 10.0i)T \) |
| good | 3 | \( 1 + (0.938 - 0.627i)T + (3.44 - 8.31i)T^{2} \) |
| 5 | \( 1 + (4.64 + 0.924i)T + (23.0 + 9.56i)T^{2} \) |
| 7 | \( 1 + (-0.473 + 0.0942i)T + (45.2 - 18.7i)T^{2} \) |
| 11 | \( 1 + (11.0 - 16.5i)T + (-46.3 - 111. i)T^{2} \) |
| 13 | \( 1 + (13.5 + 13.5i)T + 169iT^{2} \) |
| 19 | \( 1 + (-1.11 - 2.69i)T + (-255. + 255. i)T^{2} \) |
| 23 | \( 1 + (-31.9 - 21.3i)T + (202. + 488. i)T^{2} \) |
| 29 | \( 1 + (4.99 - 25.1i)T + (-776. - 321. i)T^{2} \) |
| 31 | \( 1 + (3.12 + 4.67i)T + (-367. + 887. i)T^{2} \) |
| 37 | \( 1 + (-15.5 + 10.4i)T + (523. - 1.26e3i)T^{2} \) |
| 41 | \( 1 + (-62.0 + 12.3i)T + (1.55e3 - 643. i)T^{2} \) |
| 43 | \( 1 + (-0.728 + 1.75i)T + (-1.30e3 - 1.30e3i)T^{2} \) |
| 47 | \( 1 + (-24.4 - 24.4i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (3.94 + 9.52i)T + (-1.98e3 + 1.98e3i)T^{2} \) |
| 59 | \( 1 + (75.2 + 31.1i)T + (2.46e3 + 2.46e3i)T^{2} \) |
| 61 | \( 1 + (2.52 + 12.7i)T + (-3.43e3 + 1.42e3i)T^{2} \) |
| 67 | \( 1 - 37.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (82.2 - 54.9i)T + (1.92e3 - 4.65e3i)T^{2} \) |
| 73 | \( 1 + (-6.43 - 1.28i)T + (4.92e3 + 2.03e3i)T^{2} \) |
| 79 | \( 1 + (-39.7 + 59.5i)T + (-2.38e3 - 5.76e3i)T^{2} \) |
| 83 | \( 1 + (120. - 49.8i)T + (4.87e3 - 4.87e3i)T^{2} \) |
| 89 | \( 1 + (-50.4 + 50.4i)T - 7.92e3iT^{2} \) |
| 97 | \( 1 + (25.6 - 129. i)T + (-8.69e3 - 3.60e3i)T^{2} \) |
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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
−13.12943828582340233820525383797, −12.46196090248179453006459260868, −11.32011842605831027762386371973, −10.49943793177646403827521241821, −9.365778580780269606781031763832, −7.78422297274623100286921103781, −7.38058156789036346218621458725, −5.29091014801691677769071974563, −4.58923544535841477448595337245, −2.63802626648710972249074778604,
0.21718947020573817938451672432, 2.93239171289470242847613736602, 4.42893588037765765094787421990, 5.96137598090292801274623679942, 7.08867741568183304213747274772, 8.251921184462573872195627775039, 9.251997053972683752793664624219, 10.88074722939698173887389364968, 11.42344015352041994738965373645, 12.40777863769346873221089322370
