L(s) = 1 | − 3.65i·2-s − 9.33·4-s − 5.47·5-s + 7.16i·7-s + 19.4i·8-s + 19.9i·10-s + 6.59i·13-s + 26.1·14-s + 33.7·16-s − 18.4i·17-s + 8.42i·19-s + 51.0·20-s + 17.7·23-s + 4.92·25-s + 24.0·26-s + ⋯ |
L(s) = 1 | − 1.82i·2-s − 2.33·4-s − 1.09·5-s + 1.02i·7-s + 2.43i·8-s + 1.99i·10-s + 0.507i·13-s + 1.86·14-s + 2.10·16-s − 1.08i·17-s + 0.443i·19-s + 2.55·20-s + 0.770·23-s + 0.196·25-s + 0.926·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 + 0.219i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.975 + 0.219i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8574689896\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8574689896\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 3.65iT - 4T^{2} \) |
| 5 | \( 1 + 5.47T + 25T^{2} \) |
| 7 | \( 1 - 7.16iT - 49T^{2} \) |
| 13 | \( 1 - 6.59iT - 169T^{2} \) |
| 17 | \( 1 + 18.4iT - 289T^{2} \) |
| 19 | \( 1 - 8.42iT - 361T^{2} \) |
| 23 | \( 1 - 17.7T + 529T^{2} \) |
| 29 | \( 1 - 20.0iT - 841T^{2} \) |
| 31 | \( 1 - 35.3T + 961T^{2} \) |
| 37 | \( 1 + 48.8T + 1.36e3T^{2} \) |
| 41 | \( 1 + 31.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 45.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 0.728T + 2.20e3T^{2} \) |
| 53 | \( 1 + 70.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + 24.1T + 3.48e3T^{2} \) |
| 61 | \( 1 + 41.0iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 96.0T + 4.48e3T^{2} \) |
| 71 | \( 1 + 39.5T + 5.04e3T^{2} \) |
| 73 | \( 1 + 70.2iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 89.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 26.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 118.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 33.1T + 9.40e3T^{2} \) |
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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
−9.329521897538520194716250142034, −8.844277257189280724812568465076, −8.033012773574817747966193028567, −6.88172691165201661731565828914, −5.36060189464722331244337505867, −4.63033974904846481248909726889, −3.62600544639881642187284162768, −2.88965414865469522475292102755, −1.83015682245037311022683633693, −0.40633361313443923992498740497,
0.792332597461235407280073036737, 3.40666890551555543038577113143, 4.26692748332980265139223770927, 4.90851002952611967251012251680, 6.10205465750575116119759213332, 6.81033043795542709920731985667, 7.62323337066906346019292680182, 8.041892231841753056248325834027, 8.772359333094454224465163663058, 9.842897290504097404272901281835