| L(s) = 1 | + (0.0173 − 1.99i)2-s + (3.90 − 3.20i)3-s + (−3.99 − 0.0694i)4-s + (4.26 + 1.29i)5-s + (−6.33 − 7.86i)6-s + (−0.713 − 3.58i)7-s + (−0.208 + 7.99i)8-s + (3.21 − 16.1i)9-s + (2.65 − 8.49i)10-s + (1.43 + 14.5i)11-s + (−15.8 + 12.5i)12-s + (−3.73 − 12.3i)13-s + (−7.18 + 1.36i)14-s + (20.7 − 8.60i)15-s + (15.9 + 0.555i)16-s + (−7.23 + 17.4i)17-s + ⋯ |
| L(s) = 1 | + (0.00868 − 0.999i)2-s + (1.30 − 1.06i)3-s + (−0.999 − 0.0173i)4-s + (0.852 + 0.258i)5-s + (−1.05 − 1.31i)6-s + (−0.101 − 0.512i)7-s + (−0.0260 + 0.999i)8-s + (0.357 − 1.79i)9-s + (0.265 − 0.849i)10-s + (0.130 + 1.32i)11-s + (−1.31 + 1.04i)12-s + (−0.287 − 0.946i)13-s + (−0.513 + 0.0975i)14-s + (1.38 − 0.573i)15-s + (0.999 + 0.0347i)16-s + (−0.425 + 1.02i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.481 + 0.876i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.481 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.03637 - 1.75220i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.03637 - 1.75220i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.0173 + 1.99i)T \) |
| good | 3 | \( 1 + (-3.90 + 3.20i)T + (1.75 - 8.82i)T^{2} \) |
| 5 | \( 1 + (-4.26 - 1.29i)T + (20.7 + 13.8i)T^{2} \) |
| 7 | \( 1 + (0.713 + 3.58i)T + (-45.2 + 18.7i)T^{2} \) |
| 11 | \( 1 + (-1.43 - 14.5i)T + (-118. + 23.6i)T^{2} \) |
| 13 | \( 1 + (3.73 + 12.3i)T + (-140. + 93.8i)T^{2} \) |
| 17 | \( 1 + (7.23 - 17.4i)T + (-204. - 204. i)T^{2} \) |
| 19 | \( 1 + (15.3 - 28.6i)T + (-200. - 300. i)T^{2} \) |
| 23 | \( 1 + (-12.3 + 8.22i)T + (202. - 488. i)T^{2} \) |
| 29 | \( 1 + (-4.17 + 42.4i)T + (-824. - 164. i)T^{2} \) |
| 31 | \( 1 + (-10.9 - 10.9i)T + 961iT^{2} \) |
| 37 | \( 1 + (-20.5 - 38.4i)T + (-760. + 1.13e3i)T^{2} \) |
| 41 | \( 1 + (-28.2 + 18.8i)T + (643. - 1.55e3i)T^{2} \) |
| 43 | \( 1 + (17.4 + 14.3i)T + (360. + 1.81e3i)T^{2} \) |
| 47 | \( 1 + (17.1 - 41.3i)T + (-1.56e3 - 1.56e3i)T^{2} \) |
| 53 | \( 1 + (0.0279 + 0.284i)T + (-2.75e3 + 548. i)T^{2} \) |
| 59 | \( 1 + (4.27 + 1.29i)T + (2.89e3 + 1.93e3i)T^{2} \) |
| 61 | \( 1 + (-33.7 + 27.7i)T + (725. - 3.64e3i)T^{2} \) |
| 67 | \( 1 + (-45.2 - 55.1i)T + (-875. + 4.40e3i)T^{2} \) |
| 71 | \( 1 + (48.9 - 9.73i)T + (4.65e3 - 1.92e3i)T^{2} \) |
| 73 | \( 1 + (117. + 23.4i)T + (4.92e3 + 2.03e3i)T^{2} \) |
| 79 | \( 1 + (59.3 + 143. i)T + (-4.41e3 + 4.41e3i)T^{2} \) |
| 83 | \( 1 + (51.3 + 27.4i)T + (3.82e3 + 5.72e3i)T^{2} \) |
| 89 | \( 1 + (-31.6 + 47.4i)T + (-3.03e3 - 7.31e3i)T^{2} \) |
| 97 | \( 1 + (133. - 133. i)T - 9.40e3iT^{2} \) |
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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
−12.91793470870782911769595028491, −12.17747425098326731354616396428, −10.33695815132422914492395998635, −9.846517350851996299218503849586, −8.525889408398224086086384452526, −7.65910640690075627993038274924, −6.21717685442405090393673988527, −4.11994502997313616331962817666, −2.59968163341339226583703433186, −1.63919597906157755470774580950,
2.81544044814404790444509389633, 4.37493578134526062511905233902, 5.48374877967307333594622082388, 6.94420125013003449057182128979, 8.572566057378881131686249683680, 9.073542152062236842226950262941, 9.632822684047523471708716893346, 11.12522314361138727687064051471, 13.12006542700107844543400792352, 13.77418160953777849246370824515
