| L(s) = 1 | + (−0.953 − 1.04i)2-s + (−0.132 − 0.924i)3-s + (−0.180 + 1.99i)4-s + (0.959 + 0.281i)5-s + (−0.838 + 1.02i)6-s + (−2.80 + 3.23i)7-s + (2.25 − 1.71i)8-s + (2.04 − 0.599i)9-s + (−0.620 − 1.27i)10-s + (−3.09 − 4.81i)11-s + (1.86 − 0.0976i)12-s + (−2.73 + 2.37i)13-s + (6.05 − 0.158i)14-s + (0.132 − 0.924i)15-s + (−3.93 − 0.719i)16-s + (4.43 + 2.02i)17-s + ⋯ |
| L(s) = 1 | + (−0.674 − 0.738i)2-s + (−0.0767 − 0.533i)3-s + (−0.0903 + 0.995i)4-s + (0.429 + 0.125i)5-s + (−0.342 + 0.416i)6-s + (−1.06 + 1.22i)7-s + (0.796 − 0.604i)8-s + (0.680 − 0.199i)9-s + (−0.196 − 0.401i)10-s + (−0.932 − 1.45i)11-s + (0.538 − 0.0281i)12-s + (−0.759 + 0.658i)13-s + (1.61 − 0.0423i)14-s + (0.0343 − 0.238i)15-s + (−0.983 − 0.179i)16-s + (1.07 + 0.491i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.698 - 0.715i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.698 - 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0161255 + 0.0382716i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0161255 + 0.0382716i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.953 + 1.04i)T \) |
| 5 | \( 1 + (-0.959 - 0.281i)T \) |
| 23 | \( 1 + (-1.78 + 4.45i)T \) |
| good | 3 | \( 1 + (0.132 + 0.924i)T + (-2.87 + 0.845i)T^{2} \) |
| 7 | \( 1 + (2.80 - 3.23i)T + (-0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (3.09 + 4.81i)T + (-4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (2.73 - 2.37i)T + (1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-4.43 - 2.02i)T + (11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (3.44 - 1.57i)T + (12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (2.94 + 1.34i)T + (18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (4.96 + 0.713i)T + (29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (6.20 - 1.82i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (11.1 + 3.28i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (4.00 - 0.576i)T + (41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 + 2.40iT - 47T^{2} \) |
| 53 | \( 1 + (6.44 - 7.43i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (3.78 + 4.37i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-0.158 + 1.10i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (4.57 - 7.12i)T + (-27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (-1.73 + 2.70i)T + (-29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-5.43 - 11.8i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (-10.6 - 12.2i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (-1.16 - 3.97i)T + (-69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (4.72 - 0.679i)T + (85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-4.85 + 16.5i)T + (-81.6 - 52.4i)T^{2} \) |
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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
−9.684422109349603891224499702519, −8.776489672350292563647799260512, −8.158753050978481598469188279083, −7.01291770836995611002717023008, −6.26125049499449827618016272242, −5.29446882944779366480058945754, −3.65111334958180189266182488069, −2.76481062849176391703667447122, −1.80244441716168234881184424695, −0.02344203063119919550733860562,
1.75886920897211756155675760108, 3.44299571716030236859215467534, 4.83410357302313989419450942081, 5.21514255077223242076529913606, 6.64011979866979533885715156764, 7.33491023856593553581139821795, 7.73605354641965336298727484964, 9.313793026682141402320531243169, 9.774265740497411676141608164952, 10.31805388356876031102560043286
