| L(s) = 1 | + (0.974 − 1.02i)2-s + (1.67 + 0.457i)3-s + (−0.100 − 1.99i)4-s + (−1.52 − 1.81i)5-s + (2.09 − 1.26i)6-s + (0.767 − 0.279i)7-s + (−2.14 − 1.84i)8-s + (2.58 + 1.52i)9-s + (−3.34 − 0.208i)10-s + (−3.58 + 4.26i)11-s + (0.747 − 3.38i)12-s + (2.36 − 0.417i)13-s + (0.461 − 1.05i)14-s + (−1.71 − 3.72i)15-s + (−3.97 + 0.399i)16-s + (1.52 + 2.64i)17-s + ⋯ |
| L(s) = 1 | + (0.689 − 0.724i)2-s + (0.964 + 0.264i)3-s + (−0.0500 − 0.998i)4-s + (−0.681 − 0.811i)5-s + (0.856 − 0.516i)6-s + (0.289 − 0.105i)7-s + (−0.758 − 0.652i)8-s + (0.860 + 0.509i)9-s + (−1.05 − 0.0659i)10-s + (−1.08 + 1.28i)11-s + (0.215 − 0.976i)12-s + (0.656 − 0.115i)13-s + (0.123 − 0.282i)14-s + (−0.442 − 0.962i)15-s + (−0.994 + 0.0999i)16-s + (0.370 + 0.642i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.341 + 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.341 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.64667 - 1.15351i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.64667 - 1.15351i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.974 + 1.02i)T \) |
| 3 | \( 1 + (-1.67 - 0.457i)T \) |
| good | 5 | \( 1 + (1.52 + 1.81i)T + (-0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (-0.767 + 0.279i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (3.58 - 4.26i)T + (-1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-2.36 + 0.417i)T + (12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-1.52 - 2.64i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.631 - 0.364i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-6.38 - 2.32i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (9.24 + 1.62i)T + (27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-1.92 - 0.701i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (4.61 - 2.66i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.29 + 7.32i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-4.42 + 5.26i)T + (-7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (10.2 - 3.72i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + 2.99iT - 53T^{2} \) |
| 59 | \( 1 + (0.837 + 0.998i)T + (-10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-0.528 - 1.45i)T + (-46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-4.70 + 0.829i)T + (62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (4.37 + 7.57i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.62 - 4.54i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.739 - 4.19i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-14.1 - 2.49i)T + (77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (-4.96 + 8.60i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (11.8 + 9.96i)T + (16.8 + 95.5i)T^{2} \) |
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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
−12.49617076334410575650007547044, −11.17707643819756380917964020065, −10.26107257730254398315095120539, −9.309780348855436391835711568871, −8.266836322036998384316610370216, −7.26826251653409368907905642985, −5.28954959964202673066290383150, −4.43953992934857115558249116146, −3.37434252671305017204990796462, −1.77642633302074455090707160478,
2.90074227906202400511911504337, 3.56491759589651334135787053229, 5.16386198710044329125235241791, 6.55548169947078301381411813376, 7.54685955393152043152553899434, 8.177662590782670771443016865434, 9.145640267245479904913498615107, 10.87381509334151062249307483972, 11.58174640865271283124993409940, 13.00024901771111953250994535251
