| L(s) = 1 | + (−0.293 + 0.748i)2-s + (0.992 + 0.920i)4-s + (1.11 + 0.762i)5-s + (1.80 + 1.93i)7-s + (−2.42 + 1.16i)8-s + (−0.899 + 0.613i)10-s + (−1.27 + 0.192i)11-s + (−0.117 − 0.146i)13-s + (−1.97 + 0.779i)14-s + (0.0402 + 0.537i)16-s + (2.86 + 0.882i)17-s + (−2.34 − 4.06i)19-s + (0.407 + 1.78i)20-s + (0.230 − 1.01i)22-s + (2.09 − 0.645i)23-s + ⋯ |
| L(s) = 1 | + (−0.207 + 0.529i)2-s + (0.496 + 0.460i)4-s + (0.500 + 0.340i)5-s + (0.680 + 0.732i)7-s + (−0.858 + 0.413i)8-s + (−0.284 + 0.193i)10-s + (−0.384 + 0.0579i)11-s + (−0.0324 − 0.0406i)13-s + (−0.529 + 0.208i)14-s + (0.0100 + 0.134i)16-s + (0.693 + 0.213i)17-s + (−0.538 − 0.933i)19-s + (0.0911 + 0.399i)20-s + (0.0491 − 0.215i)22-s + (0.436 − 0.134i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.203 - 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.203 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.974291 + 1.19772i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.974291 + 1.19772i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + (-1.80 - 1.93i)T \) |
| good | 2 | \( 1 + (0.293 - 0.748i)T + (-1.46 - 1.36i)T^{2} \) |
| 5 | \( 1 + (-1.11 - 0.762i)T + (1.82 + 4.65i)T^{2} \) |
| 11 | \( 1 + (1.27 - 0.192i)T + (10.5 - 3.24i)T^{2} \) |
| 13 | \( 1 + (0.117 + 0.146i)T + (-2.89 + 12.6i)T^{2} \) |
| 17 | \( 1 + (-2.86 - 0.882i)T + (14.0 + 9.57i)T^{2} \) |
| 19 | \( 1 + (2.34 + 4.06i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.09 + 0.645i)T + (19.0 - 12.9i)T^{2} \) |
| 29 | \( 1 + (-1.11 - 4.90i)T + (-26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (1.06 - 1.83i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.51 - 3.26i)T + (2.76 - 36.8i)T^{2} \) |
| 41 | \( 1 + (-9.47 + 4.56i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (-1.77 - 0.853i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (2.97 - 7.58i)T + (-34.4 - 31.9i)T^{2} \) |
| 53 | \( 1 + (5.78 + 5.36i)T + (3.96 + 52.8i)T^{2} \) |
| 59 | \( 1 + (2.53 - 1.72i)T + (21.5 - 54.9i)T^{2} \) |
| 61 | \( 1 + (-8.23 + 7.63i)T + (4.55 - 60.8i)T^{2} \) |
| 67 | \( 1 + (-6.34 + 10.9i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.310 + 1.36i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-4.84 - 12.3i)T + (-53.5 + 49.6i)T^{2} \) |
| 79 | \( 1 + (6.97 + 12.0i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-10.7 + 13.4i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (2.26 + 0.340i)T + (85.0 + 26.2i)T^{2} \) |
| 97 | \( 1 - 14.5T + 97T^{2} \) |
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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
−11.27322945613132213077503189129, −10.62615536171886864288591145838, −9.359138231484120497498257076027, −8.511137486098174134702871873692, −7.74215310129173948253263253493, −6.73893398287230521218991438535, −5.88695467377647010243861280519, −4.87244863737365877203818067395, −3.10904924906454323671637366482, −2.10955918974275699381091098338,
1.11656022178125932404626893470, 2.28367100491352576213334221806, 3.79664250255725538142832999772, 5.19653412102265011224729290518, 6.02528810908208256240508682372, 7.23307757967842470471056459856, 8.140151013648605799136275623026, 9.374159655948208635476303960174, 10.07931819898960667366760549035, 10.82075334029174205047690733924
