L(s) = 1 | + (0.789 + 0.789i)2-s + (0.548 + 1.64i)3-s − 0.754i·4-s + (−0.863 + 1.72i)6-s + (−1.97 − 1.97i)7-s + (2.17 − 2.17i)8-s + (−2.39 + 1.80i)9-s + (3.56 + 3.56i)11-s + (1.23 − 0.413i)12-s + (1.26 + 3.37i)13-s − 3.12i·14-s + 1.92·16-s + 0.700i·17-s + (−3.31 − 0.470i)18-s + (4.32 + 4.32i)19-s + ⋯ |
L(s) = 1 | + (0.558 + 0.558i)2-s + (0.316 + 0.948i)3-s − 0.377i·4-s + (−0.352 + 0.706i)6-s + (−0.747 − 0.747i)7-s + (0.768 − 0.768i)8-s + (−0.799 + 0.600i)9-s + (1.07 + 1.07i)11-s + (0.357 − 0.119i)12-s + (0.350 + 0.936i)13-s − 0.834i·14-s + 0.480·16-s + 0.169i·17-s + (−0.781 − 0.110i)18-s + (0.992 + 0.992i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0775 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0775 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.74494 + 1.61446i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.74494 + 1.61446i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.548 - 1.64i)T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (-1.26 - 3.37i)T \) |
good | 2 | \( 1 + (-0.789 - 0.789i)T + 2iT^{2} \) |
| 7 | \( 1 + (1.97 + 1.97i)T + 7iT^{2} \) |
| 11 | \( 1 + (-3.56 - 3.56i)T + 11iT^{2} \) |
| 17 | \( 1 - 0.700iT - 17T^{2} \) |
| 19 | \( 1 + (-4.32 - 4.32i)T + 19iT^{2} \) |
| 23 | \( 1 + 4.02iT - 23T^{2} \) |
| 29 | \( 1 - 6.96iT - 29T^{2} \) |
| 31 | \( 1 + (-6.53 - 6.53i)T + 31iT^{2} \) |
| 37 | \( 1 + (-2.65 - 2.65i)T + 37iT^{2} \) |
| 41 | \( 1 + (-5.37 + 5.37i)T - 41iT^{2} \) |
| 43 | \( 1 - 1.62T + 43T^{2} \) |
| 47 | \( 1 + (-3.82 + 3.82i)T - 47iT^{2} \) |
| 53 | \( 1 + 0.258T + 53T^{2} \) |
| 59 | \( 1 + (9.68 + 9.68i)T + 59iT^{2} \) |
| 61 | \( 1 + 3.98T + 61T^{2} \) |
| 67 | \( 1 + (4.24 - 4.24i)T - 67iT^{2} \) |
| 71 | \( 1 + (2.54 - 2.54i)T - 71iT^{2} \) |
| 73 | \( 1 + (6.24 + 6.24i)T + 73iT^{2} \) |
| 79 | \( 1 + 0.921T + 79T^{2} \) |
| 83 | \( 1 + (8.89 + 8.89i)T + 83iT^{2} \) |
| 89 | \( 1 + (9.13 + 9.13i)T + 89iT^{2} \) |
| 97 | \( 1 + (-1.54 + 1.54i)T - 97iT^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.12631944459611243864301639392, −9.514692025683146348759505301748, −8.751625890404181199589780025093, −7.38138528031872492209856927254, −6.72088911883192856048410609132, −5.92471535761972734786027618270, −4.72356843884385277894025968470, −4.19793565209305797520395905827, −3.33881347672401187139071244258, −1.50654095383782026848307852094,
1.02067216269233704394535975273, 2.70906124337503027520747872265, 3.04532275893910022783794812624, 4.16106928055789804295403454221, 5.73360458701811166274480139290, 6.16370770296084361362461001115, 7.41476701588479968447643657471, 8.070129450732689936920009270547, 8.972582262947460761610369065558, 9.592901824557707232619594883245