L(s) = 1 | + (0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (−0.499 − 0.866i)6-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (−1.5 − 0.866i)11-s − 0.999·12-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s − 0.999·18-s + 19-s + (−1.5 + 0.866i)22-s + (−0.5 + 0.866i)24-s + (0.5 − 0.866i)25-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (−0.499 − 0.866i)6-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (−1.5 − 0.866i)11-s − 0.999·12-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s − 0.999·18-s + 19-s + (−1.5 + 0.866i)22-s + (−0.5 + 0.866i)24-s + (0.5 − 0.866i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.330764494\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.330764494\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 17 | \( 1 + (-0.5 - 0.866i)T \) |
good | 5 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 - T + T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - 1.73iT - T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + 2T + T^{2} \) |
| 97 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
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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
−9.657081833409452974275383311449, −8.696175250606692743833760545421, −8.077026482147294199473977754099, −7.14674011120360714065045030693, −5.88843682865341629202048234177, −5.49758698073471051624411872549, −4.08454538418094095960120201721, −3.04877969705863651888985733598, −2.40260550696122733089713635539, −0.970552380087739551177582044779,
2.60824952151988611308635534859, 3.31584279792140185745499923870, 4.55986771174167643101311977389, 5.09246327245717887688869733966, 5.83143948771635030727323118306, 7.37549732525985546456343849002, 7.57391553044069187519358830980, 8.563453778224343863339620169186, 9.480359465453727802098603209391, 9.951006151419235739758435617295