L(s) = 1 | + (0.268 − 1.71i)3-s + (−0.615 + 2.57i)7-s + (−2.85 − 0.919i)9-s + (−1.80 + 1.04i)11-s − 0.245·13-s + (0.816 − 0.471i)17-s + (0.465 + 0.268i)19-s + (4.23 + 1.74i)21-s + (1.38 − 2.40i)23-s + (−2.34 + 4.63i)27-s − 0.267i·29-s + (0.981 − 0.566i)31-s + (1.29 + 3.37i)33-s + (5.33 + 3.08i)37-s + (−0.0660 + 0.420i)39-s + ⋯ |
L(s) = 1 | + (0.155 − 0.987i)3-s + (−0.232 + 0.972i)7-s + (−0.951 − 0.306i)9-s + (−0.544 + 0.314i)11-s − 0.0681·13-s + (0.198 − 0.114i)17-s + (0.106 + 0.0616i)19-s + (0.924 + 0.380i)21-s + (0.288 − 0.500i)23-s + (−0.450 + 0.892i)27-s − 0.0496i·29-s + (0.176 − 0.101i)31-s + (0.226 + 0.586i)33-s + (0.877 + 0.506i)37-s + (−0.0105 + 0.0673i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.830 - 0.556i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.830 - 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.363780454\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.363780454\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.268 + 1.71i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.615 - 2.57i)T \) |
good | 11 | \( 1 + (1.80 - 1.04i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 0.245T + 13T^{2} \) |
| 17 | \( 1 + (-0.816 + 0.471i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.465 - 0.268i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.38 + 2.40i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 0.267iT - 29T^{2} \) |
| 31 | \( 1 + (-0.981 + 0.566i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.33 - 3.08i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 2.38T + 41T^{2} \) |
| 43 | \( 1 - 11.4iT - 43T^{2} \) |
| 47 | \( 1 + (-10.7 - 6.23i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.26 - 10.8i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.25 - 10.8i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.96 - 2.86i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4.81 - 2.78i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 10.1iT - 71T^{2} \) |
| 73 | \( 1 + (-6.56 - 11.3i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.17 - 5.49i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 1.06iT - 83T^{2} \) |
| 89 | \( 1 + (-0.463 + 0.803i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 3.01T + 97T^{2} \) |
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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
−9.051500920008923645150435879687, −8.330126758706469881869458329774, −7.65751295843919884632381264339, −6.87655332949934636261829020062, −6.02176370494716866861299939044, −5.45781465786547570500545305697, −4.34061760611212062625793017806, −2.88199065724407845627626235992, −2.48074491551869442569644644196, −1.13812634738545084340443505232,
0.52288337189313621413686337925, 2.30232863467395176332745866233, 3.45272822519654759004452997183, 3.95522585663364939426068604542, 5.02109739252983934577427098741, 5.62643094062154014775120928401, 6.73732153125621732577298856396, 7.55765809517606281691731864607, 8.345126460845817944796691909402, 9.098024400123183762375821715312