| L(s) = 1 | + (−1.22 + 0.699i)2-s + (0.755 − 0.654i)3-s + (1.02 − 1.71i)4-s + (−3.85 + 0.554i)5-s + (−0.470 + 1.33i)6-s + (−1.25 + 2.74i)7-s + (−0.0506 + 2.82i)8-s + (0.142 − 0.989i)9-s + (4.34 − 3.37i)10-s + (1.60 − 5.46i)11-s + (−0.355 − 1.96i)12-s + (3.38 − 1.54i)13-s + (−0.380 − 4.25i)14-s + (−2.55 + 2.94i)15-s + (−1.91 − 3.51i)16-s + (1.64 + 1.05i)17-s + ⋯ |
| L(s) = 1 | + (−0.868 + 0.494i)2-s + (0.436 − 0.378i)3-s + (0.510 − 0.859i)4-s + (−1.72 + 0.247i)5-s + (−0.192 + 0.544i)6-s + (−0.474 + 1.03i)7-s + (−0.0179 + 0.999i)8-s + (0.0474 − 0.329i)9-s + (1.37 − 1.06i)10-s + (0.483 − 1.64i)11-s + (−0.102 − 0.568i)12-s + (0.938 − 0.428i)13-s + (−0.101 − 1.13i)14-s + (−0.658 + 0.759i)15-s + (−0.479 − 0.877i)16-s + (0.398 + 0.256i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.185i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 - 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.809909 + 0.0758939i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.809909 + 0.0758939i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.22 - 0.699i)T \) |
| 3 | \( 1 + (-0.755 + 0.654i)T \) |
| 23 | \( 1 + (1.61 - 4.51i)T \) |
| good | 5 | \( 1 + (3.85 - 0.554i)T + (4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (1.25 - 2.74i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (-1.60 + 5.46i)T + (-9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (-3.38 + 1.54i)T + (8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (-1.64 - 1.05i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (-3.00 - 4.67i)T + (-7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-1.71 + 2.66i)T + (-12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-3.99 + 4.61i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (-7.51 - 1.08i)T + (35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (-0.522 - 3.63i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (-1.49 + 1.29i)T + (6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 - 7.25T + 47T^{2} \) |
| 53 | \( 1 + (0.107 + 0.0490i)T + (34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (-9.04 + 4.13i)T + (38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (5.86 + 5.08i)T + (8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (1.13 + 3.85i)T + (-56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (-11.6 + 3.41i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (7.29 - 4.68i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (6.89 + 15.1i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (13.4 + 1.93i)T + (79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-6.52 - 7.53i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (-1.36 - 9.50i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−10.91329018542422990556033025154, −9.663313371003011281335187125095, −8.700332527398259208826763887148, −8.119607528043712817417813235588, −7.65966668369806087711350527519, −6.27761683512859151539534808878, −5.78674571993569216300316832373, −3.77762735847166782332095707538, −2.95800146398764763213645411449, −0.887813108018264529684092978282,
0.948534488521475680994274485354, 2.92449171873599466081788678308, 4.07933611562326670698287527910, 4.37342897633610721801952439006, 7.00288643054624029081584063446, 7.18057650111968328590729899736, 8.231056411465241414624454011185, 8.998393434863243243336728888658, 9.883568573546551845030207598669, 10.68025823688323483438450042966
