| L(s) = 1 | + (0.797 − 1.16i)2-s + (−0.894 − 1.33i)3-s + (−0.729 − 1.86i)4-s + (0.631 + 3.17i)5-s + (−2.27 − 0.0221i)6-s + (−0.127 − 0.306i)7-s + (−2.75 − 0.632i)8-s + (0.156 − 0.378i)9-s + (4.21 + 1.79i)10-s + (3.52 + 2.35i)11-s + (−1.84 + 2.64i)12-s + (−0.690 + 3.47i)13-s + (−0.459 − 0.0961i)14-s + (3.68 − 3.68i)15-s + (−2.93 + 2.71i)16-s + (−2.19 − 2.19i)17-s + ⋯ |
| L(s) = 1 | + (0.563 − 0.826i)2-s + (−0.516 − 0.772i)3-s + (−0.364 − 0.931i)4-s + (0.282 + 1.41i)5-s + (−0.929 − 0.00903i)6-s + (−0.0480 − 0.116i)7-s + (−0.974 − 0.223i)8-s + (0.0522 − 0.126i)9-s + (1.33 + 0.566i)10-s + (1.06 + 0.710i)11-s + (−0.531 + 0.762i)12-s + (−0.191 + 0.963i)13-s + (−0.122 − 0.0256i)14-s + (0.951 − 0.951i)15-s + (−0.734 + 0.679i)16-s + (−0.531 − 0.531i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.234 + 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.234 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.785349 - 0.618745i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.785349 - 0.618745i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.797 + 1.16i)T \) |
| good | 3 | \( 1 + (0.894 + 1.33i)T + (-1.14 + 2.77i)T^{2} \) |
| 5 | \( 1 + (-0.631 - 3.17i)T + (-4.61 + 1.91i)T^{2} \) |
| 7 | \( 1 + (0.127 + 0.306i)T + (-4.94 + 4.94i)T^{2} \) |
| 11 | \( 1 + (-3.52 - 2.35i)T + (4.20 + 10.1i)T^{2} \) |
| 13 | \( 1 + (0.690 - 3.47i)T + (-12.0 - 4.97i)T^{2} \) |
| 17 | \( 1 + (2.19 + 2.19i)T + 17iT^{2} \) |
| 19 | \( 1 + (6.74 + 1.34i)T + (17.5 + 7.27i)T^{2} \) |
| 23 | \( 1 + (0.672 + 0.278i)T + (16.2 + 16.2i)T^{2} \) |
| 29 | \( 1 + (-7.95 + 5.31i)T + (11.0 - 26.7i)T^{2} \) |
| 31 | \( 1 + 0.880iT - 31T^{2} \) |
| 37 | \( 1 + (5.44 - 1.08i)T + (34.1 - 14.1i)T^{2} \) |
| 41 | \( 1 + (-3.05 - 1.26i)T + (28.9 + 28.9i)T^{2} \) |
| 43 | \( 1 + (-1.59 + 2.38i)T + (-16.4 - 39.7i)T^{2} \) |
| 47 | \( 1 + (-3.23 - 3.23i)T + 47iT^{2} \) |
| 53 | \( 1 + (7.45 + 4.98i)T + (20.2 + 48.9i)T^{2} \) |
| 59 | \( 1 + (-0.795 - 3.99i)T + (-54.5 + 22.5i)T^{2} \) |
| 61 | \( 1 + (-2.62 - 3.92i)T + (-23.3 + 56.3i)T^{2} \) |
| 67 | \( 1 + (3.03 + 4.54i)T + (-25.6 + 61.8i)T^{2} \) |
| 71 | \( 1 + (-2.69 - 6.50i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (-4.10 + 9.91i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (1.54 - 1.54i)T - 79iT^{2} \) |
| 83 | \( 1 + (-14.5 - 2.89i)T + (76.6 + 31.7i)T^{2} \) |
| 89 | \( 1 + (1.64 - 0.679i)T + (62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 - 2.43iT - 97T^{2} \) |
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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
−14.44270246353677756545628866420, −13.58003631507389928890693456221, −12.27662157069489877788749232066, −11.55111294513324887145034483338, −10.48876981034590922586307948692, −9.318077013123285787482101920341, −6.76512932565563541281501152482, −6.45182494169950406040742421613, −4.19653919993674280550097044936, −2.20947340889011374350174719108,
4.12546774751672714710181406343, 5.15675297386782629712119833777, 6.24538646757300319781415669701, 8.286878358319551633412162234337, 9.085119722608036210124142813013, 10.66644952569460406000242018096, 12.22793563279850219545423835676, 12.95559105535265984521385139900, 14.15711884710982696402478758471, 15.44965960401862616321793095664
