L(s) = 1 | + 0.529·2-s − 1.71·4-s − 1.77·5-s − 7-s − 1.96·8-s − 0.941·10-s + 6.49·11-s − 13-s − 0.529·14-s + 2.39·16-s + 2.94·17-s − 4.83·19-s + 3.05·20-s + 3.43·22-s + 5.77·23-s − 1.83·25-s − 0.529·26-s + 1.71·28-s + 2.83·29-s + 6.27·31-s + 5.20·32-s + 1.55·34-s + 1.77·35-s + 9.55·37-s − 2.56·38-s + 3.50·40-s + 3.05·41-s + ⋯ |
L(s) = 1 | + 0.374·2-s − 0.859·4-s − 0.795·5-s − 0.377·7-s − 0.696·8-s − 0.297·10-s + 1.95·11-s − 0.277·13-s − 0.141·14-s + 0.599·16-s + 0.713·17-s − 1.10·19-s + 0.683·20-s + 0.733·22-s + 1.20·23-s − 0.367·25-s − 0.103·26-s + 0.325·28-s + 0.526·29-s + 1.12·31-s + 0.920·32-s + 0.267·34-s + 0.300·35-s + 1.57·37-s − 0.415·38-s + 0.553·40-s + 0.477·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.297865886\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.297865886\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 - 0.529T + 2T^{2} \) |
| 5 | \( 1 + 1.77T + 5T^{2} \) |
| 11 | \( 1 - 6.49T + 11T^{2} \) |
| 17 | \( 1 - 2.94T + 17T^{2} \) |
| 19 | \( 1 + 4.83T + 19T^{2} \) |
| 23 | \( 1 - 5.77T + 23T^{2} \) |
| 29 | \( 1 - 2.83T + 29T^{2} \) |
| 31 | \( 1 - 6.27T + 31T^{2} \) |
| 37 | \( 1 - 9.55T + 37T^{2} \) |
| 41 | \( 1 - 3.05T + 41T^{2} \) |
| 43 | \( 1 - 2.71T + 43T^{2} \) |
| 47 | \( 1 - 8.71T + 47T^{2} \) |
| 53 | \( 1 + 6.39T + 53T^{2} \) |
| 59 | \( 1 + 1.55T + 59T^{2} \) |
| 61 | \( 1 - 3.88T + 61T^{2} \) |
| 67 | \( 1 - 5.67T + 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 + 15.8T + 73T^{2} \) |
| 79 | \( 1 + 1.28T + 79T^{2} \) |
| 83 | \( 1 - 2.83T + 83T^{2} \) |
| 89 | \( 1 - 7.66T + 89T^{2} \) |
| 97 | \( 1 + 17.7T + 97T^{2} \) |
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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
−10.08071010811260083128778898710, −9.276233400533379632281222782591, −8.692847865976163269901723949755, −7.72567007993820665514309222023, −6.64029075089134751173747899055, −5.87755015986375565598744781051, −4.47098126957408799841196948179, −4.07110799971110409493658284738, −3.01504981100578085105169307734, −0.917997652037523746940632797488,
0.917997652037523746940632797488, 3.01504981100578085105169307734, 4.07110799971110409493658284738, 4.47098126957408799841196948179, 5.87755015986375565598744781051, 6.64029075089134751173747899055, 7.72567007993820665514309222023, 8.692847865976163269901723949755, 9.276233400533379632281222782591, 10.08071010811260083128778898710