L(s) = 1 | + (1.00 − 1.73i)2-s + (−0.879 − 1.52i)3-s + (−1.01 − 1.76i)4-s + (−0.452 + 0.784i)5-s − 3.53·6-s + (0.237 + 2.63i)7-s − 0.0686·8-s + (−0.0471 + 0.0816i)9-s + (0.909 + 1.57i)10-s + (−0.358 − 0.620i)11-s + (−1.78 + 3.09i)12-s + 13-s + (4.82 + 2.23i)14-s + 1.59·15-s + (1.96 − 3.40i)16-s + (−1.17 − 2.03i)17-s + ⋯ |
L(s) = 1 | + (0.710 − 1.22i)2-s + (−0.507 − 0.879i)3-s + (−0.508 − 0.880i)4-s + (−0.202 + 0.350i)5-s − 1.44·6-s + (0.0898 + 0.995i)7-s − 0.0242·8-s + (−0.0157 + 0.0272i)9-s + (0.287 + 0.498i)10-s + (−0.107 − 0.187i)11-s + (−0.516 + 0.894i)12-s + 0.277·13-s + (1.28 + 0.596i)14-s + 0.411·15-s + (0.491 − 0.850i)16-s + (−0.285 − 0.494i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.361 + 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.361 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.672443 - 0.982338i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.672443 - 0.982338i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-0.237 - 2.63i)T \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 + (-1.00 + 1.73i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (0.879 + 1.52i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (0.452 - 0.784i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (0.358 + 0.620i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (1.17 + 2.03i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.31 - 5.74i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.87 - 3.25i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 3.25T + 29T^{2} \) |
| 31 | \( 1 + (0.785 + 1.36i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.60 - 4.51i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 4.92T + 41T^{2} \) |
| 43 | \( 1 + 9.43T + 43T^{2} \) |
| 47 | \( 1 + (-4.15 + 7.20i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (7.04 + 12.1i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (0.358 + 0.620i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.82 + 10.0i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.69 + 8.13i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 + (-1.73 - 3.00i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (6.50 - 11.2i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 3.54T + 83T^{2} \) |
| 89 | \( 1 + (6.02 - 10.4i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 7.43T + 97T^{2} \) |
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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
−13.31393303437901177366850286674, −12.50615895860765964737112535821, −11.80458218704909615986894830037, −11.08989564986948798115249852632, −9.764835585924940091771901797573, −8.103002994600222945331782699948, −6.59877797205958214651940996322, −5.31618080681637123914031186301, −3.51892770271674247484985365349, −1.89202839583541384462979097762,
4.24429525564441193119084043889, 4.74040104744342984958208378829, 6.20917471725666159071687681210, 7.34952472058576369051691349193, 8.611472007712786356739646341112, 10.31761220347988093153010821796, 10.97525115350448860444764131721, 12.68927234435081492920885263221, 13.60195387985080456643250353259, 14.60392738192361276560433824903