| L(s) = 1 | + (−1.37 + 1.92i)2-s + (0.235 − 0.971i)3-s + (−1.17 − 3.38i)4-s + (0.00886 − 0.186i)5-s + (1.54 + 1.78i)6-s + (0.229 + 2.63i)7-s + (3.57 + 1.05i)8-s + (−0.888 − 0.458i)9-s + (0.345 + 0.272i)10-s + (1.99 + 2.80i)11-s + (−3.56 + 0.340i)12-s + (1.00 − 6.96i)13-s + (−5.38 − 3.17i)14-s + (−0.178 − 0.0524i)15-s + (−1.29 + 1.01i)16-s + (0.821 − 0.158i)17-s + ⋯ |
| L(s) = 1 | + (−0.968 + 1.36i)2-s + (0.136 − 0.561i)3-s + (−0.585 − 1.69i)4-s + (0.00396 − 0.0832i)5-s + (0.631 + 0.728i)6-s + (0.0865 + 0.996i)7-s + (1.26 + 0.371i)8-s + (−0.296 − 0.152i)9-s + (0.109 + 0.0860i)10-s + (0.602 + 0.845i)11-s + (−1.02 + 0.0981i)12-s + (0.277 − 1.93i)13-s + (−1.43 − 0.847i)14-s + (−0.0461 − 0.0135i)15-s + (−0.323 + 0.254i)16-s + (0.199 − 0.0384i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0228 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0228 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.636632 + 0.622264i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.636632 + 0.622264i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.235 + 0.971i)T \) |
| 7 | \( 1 + (-0.229 - 2.63i)T \) |
| 23 | \( 1 + (0.101 - 4.79i)T \) |
| good | 2 | \( 1 + (1.37 - 1.92i)T + (-0.654 - 1.89i)T^{2} \) |
| 5 | \( 1 + (-0.00886 + 0.186i)T + (-4.97 - 0.475i)T^{2} \) |
| 11 | \( 1 + (-1.99 - 2.80i)T + (-3.59 + 10.3i)T^{2} \) |
| 13 | \( 1 + (-1.00 + 6.96i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (-0.821 + 0.158i)T + (15.7 - 6.31i)T^{2} \) |
| 19 | \( 1 + (-1.86 - 0.360i)T + (17.6 + 7.06i)T^{2} \) |
| 29 | \( 1 + (-4.93 - 5.69i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (7.25 - 6.91i)T + (1.47 - 30.9i)T^{2} \) |
| 37 | \( 1 + (-1.27 - 0.659i)T + (21.4 + 30.1i)T^{2} \) |
| 41 | \( 1 + (-1.71 - 1.10i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (-10.1 + 2.98i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + (-0.421 - 0.729i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-11.5 - 4.62i)T + (38.3 + 36.5i)T^{2} \) |
| 59 | \( 1 + (4.03 + 3.17i)T + (13.9 + 57.3i)T^{2} \) |
| 61 | \( 1 + (1.98 + 8.18i)T + (-54.2 + 27.9i)T^{2} \) |
| 67 | \( 1 + (-10.0 - 0.955i)T + (65.7 + 12.6i)T^{2} \) |
| 71 | \( 1 + (1.02 + 2.25i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (-2.49 - 7.21i)T + (-57.3 + 45.1i)T^{2} \) |
| 79 | \( 1 + (6.19 - 2.47i)T + (57.1 - 54.5i)T^{2} \) |
| 83 | \( 1 + (8.77 - 5.64i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (7.80 + 7.44i)T + (4.23 + 88.8i)T^{2} \) |
| 97 | \( 1 + (-4.61 - 2.96i)T + (40.2 + 88.2i)T^{2} \) |
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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
−10.94758063310608227698044474184, −9.921355653696925335125349605499, −9.039289022044758729437768554510, −8.449816101671183830362695907511, −7.55300290441636047942936107097, −6.89479102657979984952156210871, −5.73468386527234240665084899424, −5.23446842359266012440200008707, −3.07580665923873099939715887436, −1.22228883451036216761185891905,
0.929752211005862173897189866259, 2.42613106159911049145136968295, 3.78149445346913186884653054017, 4.32525138271718119466643925293, 6.23569713061594890157311347560, 7.42044770451185063133485122957, 8.597051382255441703496608498009, 9.154275088729090807694566925094, 9.936834548611471245496713132913, 10.80928867421237592568131385202
