| L(s) = 1 | + (−2.23 − 0.0974i)5-s + 1.31i·7-s + (1.25 − 0.913i)11-s + (1.42 − 1.96i)13-s + (1.25 − 0.406i)17-s + (0.315 + 0.971i)19-s + (2.94 + 4.05i)23-s + (4.98 + 0.435i)25-s + (−1.82 + 5.61i)29-s + (2.73 + 8.41i)31-s + (0.128 − 2.94i)35-s + (−2.95 + 4.06i)37-s + (6.43 + 4.67i)41-s − 6.84i·43-s + (7.37 + 2.39i)47-s + ⋯ |
| L(s) = 1 | + (−0.999 − 0.0435i)5-s + 0.498i·7-s + (0.379 − 0.275i)11-s + (0.395 − 0.544i)13-s + (0.303 − 0.0985i)17-s + (0.0723 + 0.222i)19-s + (0.613 + 0.844i)23-s + (0.996 + 0.0870i)25-s + (−0.338 + 1.04i)29-s + (0.491 + 1.51i)31-s + (0.0217 − 0.497i)35-s + (−0.485 + 0.668i)37-s + (1.00 + 0.730i)41-s − 1.04i·43-s + (1.07 + 0.349i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.785 - 0.618i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.785 - 0.618i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.22268 + 0.423465i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.22268 + 0.423465i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.23 + 0.0974i)T \) |
| good | 7 | \( 1 - 1.31iT - 7T^{2} \) |
| 11 | \( 1 + (-1.25 + 0.913i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-1.42 + 1.96i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-1.25 + 0.406i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-0.315 - 0.971i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-2.94 - 4.05i)T + (-7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (1.82 - 5.61i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-2.73 - 8.41i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (2.95 - 4.06i)T + (-11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-6.43 - 4.67i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 6.84iT - 43T^{2} \) |
| 47 | \( 1 + (-7.37 - 2.39i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-3.75 - 1.22i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-6.35 - 4.61i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-2.83 + 2.05i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (7.92 - 2.57i)T + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (-4.00 + 12.3i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (7.47 + 10.2i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-0.386 + 1.18i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-7.80 + 2.53i)T + (67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (7.74 - 5.62i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (15.4 + 5.02i)T + (78.4 + 57.0i)T^{2} \) |
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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
−10.37645585063785680183118430975, −9.089913668686541600611316945734, −8.626251987879355396498868987643, −7.66498351774940094514208775904, −6.92659267369972862041488559103, −5.78037625525080820932987366285, −4.92635397300273956773629983912, −3.72333483806414673840817591916, −2.97594723444673792094543639976, −1.16287865803270715666019054467,
0.77505552942782206554646796556, 2.51963097399977626125090977911, 3.95677389315970832163586569601, 4.29893945342341030494027889691, 5.69324293946827671511173975701, 6.79807431880391065657490046220, 7.43019896975821434481233347771, 8.301527037767621140139029754153, 9.121934871575875652682265553743, 10.05409402779296814773345676897
