| L(s) = 1 | + (−5.64 + 6.73i)5-s + (4.05 + 1.47i)7-s + (−12.6 − 15.0i)11-s + (−1.31 + 7.45i)13-s + (3.16 + 1.82i)17-s + (−16.8 − 29.1i)19-s + (−1.05 − 2.88i)23-s + (−9.07 − 51.4i)25-s + (−23.9 + 4.22i)29-s + (−27.1 + 9.87i)31-s + (−32.8 + 18.9i)35-s + (−14.9 + 25.8i)37-s + (22.6 + 3.99i)41-s + (−6.85 + 5.74i)43-s + (−3.60 + 9.90i)47-s + ⋯ |
| L(s) = 1 | + (−1.12 + 1.34i)5-s + (0.578 + 0.210i)7-s + (−1.15 − 1.37i)11-s + (−0.101 + 0.573i)13-s + (0.186 + 0.107i)17-s + (−0.885 − 1.53i)19-s + (−0.0457 − 0.125i)23-s + (−0.362 − 2.05i)25-s + (−0.825 + 0.145i)29-s + (−0.875 + 0.318i)31-s + (−0.937 + 0.541i)35-s + (−0.402 + 0.697i)37-s + (0.552 + 0.0974i)41-s + (−0.159 + 0.133i)43-s + (−0.0766 + 0.210i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 + 0.434i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.900 + 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.00966242 - 0.0422606i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00966242 - 0.0422606i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (5.64 - 6.73i)T + (-4.34 - 24.6i)T^{2} \) |
| 7 | \( 1 + (-4.05 - 1.47i)T + (37.5 + 31.4i)T^{2} \) |
| 11 | \( 1 + (12.6 + 15.0i)T + (-21.0 + 119. i)T^{2} \) |
| 13 | \( 1 + (1.31 - 7.45i)T + (-158. - 57.8i)T^{2} \) |
| 17 | \( 1 + (-3.16 - 1.82i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (16.8 + 29.1i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (1.05 + 2.88i)T + (-405. + 340. i)T^{2} \) |
| 29 | \( 1 + (23.9 - 4.22i)T + (790. - 287. i)T^{2} \) |
| 31 | \( 1 + (27.1 - 9.87i)T + (736. - 617. i)T^{2} \) |
| 37 | \( 1 + (14.9 - 25.8i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-22.6 - 3.99i)T + (1.57e3 + 574. i)T^{2} \) |
| 43 | \( 1 + (6.85 - 5.74i)T + (321. - 1.82e3i)T^{2} \) |
| 47 | \( 1 + (3.60 - 9.90i)T + (-1.69e3 - 1.41e3i)T^{2} \) |
| 53 | \( 1 - 70.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-43.8 + 52.3i)T + (-604. - 3.42e3i)T^{2} \) |
| 61 | \( 1 + (99.7 + 36.3i)T + (2.85e3 + 2.39e3i)T^{2} \) |
| 67 | \( 1 + (4.27 - 24.2i)T + (-4.21e3 - 1.53e3i)T^{2} \) |
| 71 | \( 1 + (29.9 + 17.2i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-20.9 - 36.2i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-8.78 - 49.8i)T + (-5.86e3 + 2.13e3i)T^{2} \) |
| 83 | \( 1 + (-30.9 + 5.45i)T + (6.47e3 - 2.35e3i)T^{2} \) |
| 89 | \( 1 + (40.0 - 23.1i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-84.8 + 71.1i)T + (1.63e3 - 9.26e3i)T^{2} \) |
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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
−11.48587626780328428580504014081, −11.11656626495406058407895643668, −10.48797207584195783237280344656, −8.910133823615896109817008480575, −8.035786398644338016783384916006, −7.24301911578551617644003262600, −6.22393169730152164666196768383, −4.85844733423273605894462347705, −3.53550668346459597975913548086, −2.54856525628254722983520476306,
0.01963935518173916852298440515, 1.80441237841840313149547864806, 3.83753233272945610013827473410, 4.72535967566055823273189660334, 5.56440085720264269515051212672, 7.57062262047538868454523906044, 7.79092833641866477644175299081, 8.806457487326363378491709250994, 9.970493161173074323371957726016, 10.89344751586945185681852703181
