L(s) = 1 | + 2.81i·3-s + 0.289·5-s + 4.38·7-s − 4.91·9-s + (2.56 + 2.10i)11-s + 1.55i·13-s + 0.813i·15-s − 3.57i·17-s − 6.39·19-s + 12.3i·21-s − 3.39i·23-s − 4.91·25-s − 5.39i·27-s − 3.64i·29-s + 9.01i·31-s + ⋯ |
L(s) = 1 | + 1.62i·3-s + 0.129·5-s + 1.65·7-s − 1.63·9-s + (0.773 + 0.634i)11-s + 0.432i·13-s + 0.210i·15-s − 0.865i·17-s − 1.46·19-s + 2.69i·21-s − 0.707i·23-s − 0.983·25-s − 1.03i·27-s − 0.677i·29-s + 1.61i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0985 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0985 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.03758 + 1.14537i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.03758 + 1.14537i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 + (-2.56 - 2.10i)T \) |
good | 3 | \( 1 - 2.81iT - 3T^{2} \) |
| 5 | \( 1 - 0.289T + 5T^{2} \) |
| 7 | \( 1 - 4.38T + 7T^{2} \) |
| 13 | \( 1 - 1.55iT - 13T^{2} \) |
| 17 | \( 1 + 3.57iT - 17T^{2} \) |
| 19 | \( 1 + 6.39T + 19T^{2} \) |
| 23 | \( 1 + 3.39iT - 23T^{2} \) |
| 29 | \( 1 + 3.64iT - 29T^{2} \) |
| 31 | \( 1 - 9.01iT - 31T^{2} \) |
| 37 | \( 1 - 0.289T + 37T^{2} \) |
| 41 | \( 1 + 6.68iT - 41T^{2} \) |
| 43 | \( 1 - 5.12T + 43T^{2} \) |
| 47 | \( 1 + 7.04iT - 47T^{2} \) |
| 53 | \( 1 - 9.25T + 53T^{2} \) |
| 59 | \( 1 + 0.440iT - 59T^{2} \) |
| 61 | \( 1 + 10.7iT - 61T^{2} \) |
| 67 | \( 1 - 3.97iT - 67T^{2} \) |
| 71 | \( 1 - 5.76iT - 71T^{2} \) |
| 73 | \( 1 - 6.68iT - 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 + 13.9T + 83T^{2} \) |
| 89 | \( 1 - 12.5T + 89T^{2} \) |
| 97 | \( 1 + 8.75T + 97T^{2} \) |
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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
−11.46195501714022472632297579238, −10.75747682913166603152787097346, −9.965658833956731544512601508490, −8.986484738644441079272429037293, −8.356857552820817715575818490165, −6.95462794952156757608656533788, −5.46970829096548700950721702262, −4.56689087813845391280521980125, −4.02376131674035028588013591985, −2.12569216467587796842777519458,
1.26802448062448540270173985493, 2.21352439853079373817864203079, 4.13446783422625853316052787695, 5.68121311991338657963387941031, 6.39017323270040025468311826485, 7.69147402806712418928191971745, 8.087876182012918181236422964529, 9.005302229633835186785331931143, 10.68617484013368774936382982957, 11.43464949725883986384560834663