| L(s) = 1 | + (1.34 + 0.448i)2-s + (−0.802 + 0.601i)3-s + (1.59 + 1.20i)4-s + (−2.12 − 0.681i)5-s + (−1.34 + 0.446i)6-s + (−2.03 − 0.145i)7-s + (1.60 + 2.32i)8-s + (−0.561 + 1.91i)9-s + (−2.55 − 1.86i)10-s + (−3.15 + 4.91i)11-s + (−2.00 − 0.00481i)12-s + (−0.0276 − 0.386i)13-s + (−2.66 − 1.10i)14-s + (2.11 − 0.732i)15-s + (1.10 + 3.84i)16-s + (−0.392 + 1.05i)17-s + ⋯ |
| L(s) = 1 | + (0.948 + 0.316i)2-s + (−0.463 + 0.347i)3-s + (0.799 + 0.601i)4-s + (−0.952 − 0.304i)5-s + (−0.549 + 0.182i)6-s + (−0.770 − 0.0551i)7-s + (0.567 + 0.823i)8-s + (−0.187 + 0.637i)9-s + (−0.806 − 0.590i)10-s + (−0.951 + 1.48i)11-s + (−0.579 − 0.00138i)12-s + (−0.00767 − 0.107i)13-s + (−0.713 − 0.296i)14-s + (0.547 − 0.189i)15-s + (0.277 + 0.960i)16-s + (−0.0951 + 0.255i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.780 - 0.625i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.780 - 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.413694 + 1.17717i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.413694 + 1.17717i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.34 - 0.448i)T \) |
| 5 | \( 1 + (2.12 + 0.681i)T \) |
| 23 | \( 1 + (-3.78 - 2.95i)T \) |
| good | 3 | \( 1 + (0.802 - 0.601i)T + (0.845 - 2.87i)T^{2} \) |
| 7 | \( 1 + (2.03 + 0.145i)T + (6.92 + 0.996i)T^{2} \) |
| 11 | \( 1 + (3.15 - 4.91i)T + (-4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (0.0276 + 0.386i)T + (-12.8 + 1.85i)T^{2} \) |
| 17 | \( 1 + (0.392 - 1.05i)T + (-12.8 - 11.1i)T^{2} \) |
| 19 | \( 1 + (-0.986 + 2.16i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (0.795 - 0.363i)T + (18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-4.64 + 0.668i)T + (29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (2.30 + 4.21i)T + (-20.0 + 31.1i)T^{2} \) |
| 41 | \( 1 + (6.59 - 1.93i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-0.431 + 0.323i)T + (12.1 - 41.2i)T^{2} \) |
| 47 | \( 1 + (-6.63 + 6.63i)T - 47iT^{2} \) |
| 53 | \( 1 + (-3.45 - 0.246i)T + (52.4 + 7.54i)T^{2} \) |
| 59 | \( 1 + (1.05 - 1.21i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (-0.439 - 3.05i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (-2.15 + 9.92i)T + (-60.9 - 27.8i)T^{2} \) |
| 71 | \( 1 + (-7.77 - 12.0i)T + (-29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-5.28 - 14.1i)T + (-55.1 + 47.8i)T^{2} \) |
| 79 | \( 1 + (6.31 - 7.28i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-5.75 - 10.5i)T + (-44.8 + 69.8i)T^{2} \) |
| 89 | \( 1 + (-4.87 - 0.700i)T + (85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (1.29 + 0.704i)T + (52.4 + 81.6i)T^{2} \) |
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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
−11.53229246209475546856961724975, −10.74493350394678111115932840900, −9.855974345718971372557833247309, −8.417841222878871064271322283004, −7.48509125352664119022553433484, −6.84089678322972241387328660921, −5.38043570471244149867480001886, −4.83081055627943786205324305468, −3.80146296503605026530767995563, −2.51248333499305356572629026172,
0.57989439128876601777935660446, 2.95974156852122437054645273847, 3.50634934106626531255454922189, 4.93885935754275349957861554691, 6.05473884852320181449049678888, 6.65043949297338759470645197988, 7.70975079765876167700259836925, 8.879853178845010818070568402500, 10.25348192666256833083040799864, 10.98587979251005777572218714530
