| L(s) = 1 | + (−0.405 − 1.95i)2-s + (−0.340 − 2.98i)3-s + (−3.67 + 1.58i)4-s + (−1.53 − 2.65i)5-s + (−5.69 + 1.87i)6-s + (−0.720 + 1.24i)7-s + (4.60 + 6.54i)8-s + (−8.76 + 2.03i)9-s + (−4.57 + 4.07i)10-s + (8.82 − 15.2i)11-s + (5.98 + 10.3i)12-s + (−8.00 + 4.61i)13-s + (2.73 + 0.904i)14-s + (−7.38 + 5.46i)15-s + (10.9 − 11.6i)16-s − 4.69i·17-s + ⋯ |
| L(s) = 1 | + (−0.202 − 0.979i)2-s + (−0.113 − 0.993i)3-s + (−0.917 + 0.397i)4-s + (−0.306 − 0.530i)5-s + (−0.949 + 0.312i)6-s + (−0.102 + 0.178i)7-s + (0.575 + 0.817i)8-s + (−0.974 + 0.225i)9-s + (−0.457 + 0.407i)10-s + (0.802 − 1.39i)11-s + (0.499 + 0.866i)12-s + (−0.615 + 0.355i)13-s + (0.195 + 0.0646i)14-s + (−0.492 + 0.364i)15-s + (0.684 − 0.729i)16-s − 0.276i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.142i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.989 + 0.142i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0600630 - 0.840587i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0600630 - 0.840587i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.405 + 1.95i)T \) |
| 3 | \( 1 + (0.340 + 2.98i)T \) |
| good | 5 | \( 1 + (1.53 + 2.65i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (0.720 - 1.24i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-8.82 + 15.2i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (8.00 - 4.61i)T + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + 4.69iT - 289T^{2} \) |
| 19 | \( 1 + 16.8iT - 361T^{2} \) |
| 23 | \( 1 + (-33.8 + 19.5i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (7.60 - 13.1i)T + (-420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (-24.3 - 42.1i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 14.1iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-8.78 + 5.07i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (19.3 + 11.1i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (36.8 + 21.2i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + 71.2T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-34.9 - 60.6i)T + (-1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-89.7 - 51.8i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-10.5 + 6.07i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 112. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 84.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + (22.9 - 39.8i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-25.7 + 44.6i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 - 105. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-19.4 + 33.6i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−13.56555653811979235833358631032, −12.56923667278705908144469447616, −11.77335315973620770780517713061, −10.89100083592845482011650346995, −9.046068443332917170578822552315, −8.428747024570255449356840465097, −6.78382305603913380330549004571, −4.95770733580566469963681955025, −2.94389945170877018206105305460, −0.864564527522731305563881476486,
3.83559812719838210660206503810, 5.11530119119936681314533002102, 6.65411429429175351920201054814, 7.83823359636691402493912595441, 9.413896293774256899514066113397, 10.02200720995834526687803332468, 11.39899255934142099715946511885, 12.90454851425163052001403208318, 14.54618347019441981968101212027, 14.90705475866666047890559536784
