| L(s) = 1 | + (0.902 − 1.08i)2-s + (0.220 − 0.304i)3-s + (−0.372 − 1.96i)4-s + (−0.321 − 2.21i)5-s + (−0.131 − 0.514i)6-s + (−0.601 − 0.601i)7-s + (−2.47 − 1.36i)8-s + (0.883 + 2.71i)9-s + (−2.69 − 1.64i)10-s + (−0.0459 + 0.0234i)11-s + (−0.679 − 0.321i)12-s + (−1.67 − 5.16i)13-s + (−1.19 + 0.112i)14-s + (−0.743 − 0.391i)15-s + (−3.72 + 1.46i)16-s + (−2.65 − 0.421i)17-s + ⋯ |
| L(s) = 1 | + (0.637 − 0.770i)2-s + (0.127 − 0.175i)3-s + (−0.186 − 0.982i)4-s + (−0.143 − 0.989i)5-s + (−0.0538 − 0.210i)6-s + (−0.227 − 0.227i)7-s + (−0.875 − 0.483i)8-s + (0.294 + 0.906i)9-s + (−0.853 − 0.520i)10-s + (−0.0138 + 0.00706i)11-s + (−0.196 − 0.0926i)12-s + (−0.465 − 1.43i)13-s + (−0.320 + 0.0300i)14-s + (−0.192 − 0.101i)15-s + (−0.930 + 0.365i)16-s + (−0.644 − 0.102i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.778 + 0.627i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.778 + 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.573980 - 1.62694i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.573980 - 1.62694i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.902 + 1.08i)T \) |
| 5 | \( 1 + (0.321 + 2.21i)T \) |
| good | 3 | \( 1 + (-0.220 + 0.304i)T + (-0.927 - 2.85i)T^{2} \) |
| 7 | \( 1 + (0.601 + 0.601i)T + 7iT^{2} \) |
| 11 | \( 1 + (0.0459 - 0.0234i)T + (6.46 - 8.89i)T^{2} \) |
| 13 | \( 1 + (1.67 + 5.16i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (2.65 + 0.421i)T + (16.1 + 5.25i)T^{2} \) |
| 19 | \( 1 + (-7.82 - 1.24i)T + (18.0 + 5.87i)T^{2} \) |
| 23 | \( 1 + (-1.37 - 2.69i)T + (-13.5 + 18.6i)T^{2} \) |
| 29 | \( 1 + (-4.32 + 0.684i)T + (27.5 - 8.96i)T^{2} \) |
| 31 | \( 1 + (3.47 + 4.77i)T + (-9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-0.634 - 1.95i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-3.90 + 1.26i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 1.21T + 43T^{2} \) |
| 47 | \( 1 + (-8.37 + 1.32i)T + (44.6 - 14.5i)T^{2} \) |
| 53 | \( 1 + (-2.31 + 3.18i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-5.17 - 2.63i)T + (34.6 + 47.7i)T^{2} \) |
| 61 | \( 1 + (7.52 - 3.83i)T + (35.8 - 49.3i)T^{2} \) |
| 67 | \( 1 + (0.613 - 0.445i)T + (20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (-7.50 - 5.45i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.231 + 0.117i)T + (42.9 - 59.0i)T^{2} \) |
| 79 | \( 1 + (-12.8 - 9.37i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-4.38 - 6.03i)T + (-25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (4.64 - 14.2i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (2.86 + 18.1i)T + (-92.2 + 29.9i)T^{2} \) |
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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
−11.02168273150593671039417446949, −10.07710501989124527959544925985, −9.379407803414137419024996972117, −8.153138966938990761024298713663, −7.25437601321349831111022317798, −5.57939515757100140909535576665, −5.06450175857836930954092507583, −3.85697233383250686031549650285, −2.53992542525730986806865895287, −0.969938598704341130368318076744,
2.69439827504531020890432398829, 3.71108166720604439009174665386, 4.75268373237544051685696695472, 6.15450441325909135968091057596, 6.86057241206001081059968623486, 7.51357954740162412468946967237, 8.987875812981953509063746291732, 9.500008722452566245520213701787, 10.88435605983731560952807561186, 11.88050460676595095419002097439
