| L(s) = 1 | + (2.31 − 1.33i)2-s + (0.256 − 2.98i)3-s + (1.58 − 2.74i)4-s + (1.93 − 1.11i)5-s + (−3.40 − 7.27i)6-s + (4.89 + 5.00i)7-s + 2.23i·8-s + (−8.86 − 1.53i)9-s + (2.99 − 5.18i)10-s + (−15.6 − 9.04i)11-s + (−7.78 − 5.43i)12-s + 2.70·13-s + (18.0 + 5.05i)14-s + (−2.84 − 6.07i)15-s + (9.32 + 16.1i)16-s + (3.39 + 1.96i)17-s + ⋯ |
| L(s) = 1 | + (1.15 − 0.669i)2-s + (0.0855 − 0.996i)3-s + (0.395 − 0.685i)4-s + (0.387 − 0.223i)5-s + (−0.567 − 1.21i)6-s + (0.699 + 0.714i)7-s + 0.279i·8-s + (−0.985 − 0.170i)9-s + (0.299 − 0.518i)10-s + (−1.42 − 0.822i)11-s + (−0.649 − 0.452i)12-s + 0.208·13-s + (1.28 + 0.360i)14-s + (−0.189 − 0.405i)15-s + (0.582 + 1.00i)16-s + (0.199 + 0.115i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.101 + 0.994i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.101 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.81533 - 1.63967i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.81533 - 1.63967i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.256 + 2.98i)T \) |
| 5 | \( 1 + (-1.93 + 1.11i)T \) |
| 7 | \( 1 + (-4.89 - 5.00i)T \) |
| good | 2 | \( 1 + (-2.31 + 1.33i)T + (2 - 3.46i)T^{2} \) |
| 11 | \( 1 + (15.6 + 9.04i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 - 2.70T + 169T^{2} \) |
| 17 | \( 1 + (-3.39 - 1.96i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-14.7 - 25.6i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-5.66 + 3.27i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 - 18.8iT - 841T^{2} \) |
| 31 | \( 1 + (-12.6 + 21.9i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (33.5 + 58.0i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + 38.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 63.9T + 1.84e3T^{2} \) |
| 47 | \( 1 + (33.6 - 19.4i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-0.787 - 0.454i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-20.6 - 11.9i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (25.3 + 43.8i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (34.7 - 60.1i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 55.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-14.8 + 25.7i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (14.9 + 25.9i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + 78.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-133. + 77.0i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 - 32.8T + 9.40e3T^{2} \) |
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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
−13.19011309232447354705869328120, −12.41711934084078982644329281453, −11.60332176830322428695086330116, −10.58541764304722101741610808062, −8.646010538565394179333836814151, −7.82812320883719566351297858597, −5.82981153681469095155957394792, −5.27141233689535077335931880244, −3.17836616448846624532057692152, −1.89728821755233347188819368403,
3.12004363521295084320588607684, 4.75304165787844212989924391798, 5.15879738514034848657028930830, 6.80072591822701033139832099491, 8.021668195219003689274105853926, 9.737577345194951257664924690611, 10.50434721560943399846708650294, 11.73064608401059936186305554545, 13.36071458801750244926523561247, 13.73934195476455350044808521147
