| L(s) = 1 | + (−1.40 − 0.166i)2-s + (−1.73 + 0.0744i)3-s + (1.94 + 0.466i)4-s + (−0.720 − 1.07i)5-s + (2.44 + 0.182i)6-s + (−0.767 + 1.85i)7-s + (−2.65 − 0.978i)8-s + (2.98 − 0.257i)9-s + (0.833 + 1.63i)10-s + (0.967 + 0.192i)11-s + (−3.40 − 0.662i)12-s + (2.76 + 1.84i)13-s + (1.38 − 2.47i)14-s + (1.32 + 1.81i)15-s + (3.56 + 1.81i)16-s + (4.20 + 4.20i)17-s + ⋯ |
| L(s) = 1 | + (−0.993 − 0.117i)2-s + (−0.999 + 0.0429i)3-s + (0.972 + 0.233i)4-s + (−0.322 − 0.482i)5-s + (0.997 + 0.0746i)6-s + (−0.289 + 0.699i)7-s + (−0.938 − 0.345i)8-s + (0.996 − 0.0859i)9-s + (0.263 + 0.516i)10-s + (0.291 + 0.0580i)11-s + (−0.981 − 0.191i)12-s + (0.767 + 0.512i)13-s + (0.370 − 0.661i)14-s + (0.342 + 0.468i)15-s + (0.891 + 0.453i)16-s + (1.01 + 1.01i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.817 - 0.575i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.817 - 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.513616 + 0.162559i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.513616 + 0.162559i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.40 + 0.166i)T \) |
| 3 | \( 1 + (1.73 - 0.0744i)T \) |
| good | 5 | \( 1 + (0.720 + 1.07i)T + (-1.91 + 4.61i)T^{2} \) |
| 7 | \( 1 + (0.767 - 1.85i)T + (-4.94 - 4.94i)T^{2} \) |
| 11 | \( 1 + (-0.967 - 0.192i)T + (10.1 + 4.20i)T^{2} \) |
| 13 | \( 1 + (-2.76 - 1.84i)T + (4.97 + 12.0i)T^{2} \) |
| 17 | \( 1 + (-4.20 - 4.20i)T + 17iT^{2} \) |
| 19 | \( 1 + (-1.31 - 0.879i)T + (7.27 + 17.5i)T^{2} \) |
| 23 | \( 1 + (-3.41 - 8.25i)T + (-16.2 + 16.2i)T^{2} \) |
| 29 | \( 1 + (0.303 + 1.52i)T + (-26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 - 1.85T + 31T^{2} \) |
| 37 | \( 1 + (3.53 + 5.29i)T + (-14.1 + 34.1i)T^{2} \) |
| 41 | \( 1 + (-0.606 + 0.251i)T + (28.9 - 28.9i)T^{2} \) |
| 43 | \( 1 + (7.79 + 1.54i)T + (39.7 + 16.4i)T^{2} \) |
| 47 | \( 1 + (3.56 - 3.56i)T - 47iT^{2} \) |
| 53 | \( 1 + (0.605 - 3.04i)T + (-48.9 - 20.2i)T^{2} \) |
| 59 | \( 1 + (-6.01 + 4.01i)T + (22.5 - 54.5i)T^{2} \) |
| 61 | \( 1 + (-2.31 - 11.6i)T + (-56.3 + 23.3i)T^{2} \) |
| 67 | \( 1 + (-12.2 + 2.43i)T + (61.8 - 25.6i)T^{2} \) |
| 71 | \( 1 + (10.2 + 4.23i)T + (50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (8.22 - 3.40i)T + (51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (-4.22 + 4.22i)T - 79iT^{2} \) |
| 83 | \( 1 + (-8.63 + 12.9i)T + (-31.7 - 76.6i)T^{2} \) |
| 89 | \( 1 + (-10.4 - 4.32i)T + (62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 - 4.02iT - 97T^{2} \) |
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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
−12.14307148597385510711203233265, −11.71439792788628969663136535013, −10.66529589640845641123484048717, −9.673038267739063644282297408622, −8.766264482344767399089957749556, −7.62244319453026498270847336173, −6.40457777562962491335017721622, −5.50846534419731542596944761741, −3.68844594441271233678242696048, −1.39528535189350007281672872130,
0.871277895285357529747119070940, 3.30868336867952353048361825683, 5.20497246716633059225906010002, 6.59769448952839462454151747394, 7.11056440229037908574306622427, 8.324569248994521945972548725373, 9.746558263353965763074891780409, 10.49762372699454363804785017982, 11.23629131819448250388260880384, 12.04738624619939933147403349529
