L(s) = 1 | + (−0.642 − 1.60i)3-s + (−1.28 − 2.23i)5-s + (−2.17 + 2.06i)9-s + (−1.43 − 0.826i)11-s − 5.71i·13-s + (−2.76 + 3.50i)15-s + (−3.79 + 6.56i)17-s + (−2.58 + 1.49i)19-s + (−0.249 + 0.143i)23-s + (−0.825 + 1.43i)25-s + (4.72 + 2.16i)27-s − 2.05i·29-s + (5.21 + 3.00i)31-s + (−0.409 + 2.83i)33-s + (−0.877 − 1.51i)37-s + ⋯ |
L(s) = 1 | + (−0.371 − 0.928i)3-s + (−0.576 − 0.998i)5-s + (−0.724 + 0.689i)9-s + (−0.431 − 0.249i)11-s − 1.58i·13-s + (−0.713 + 0.906i)15-s + (−0.919 + 1.59i)17-s + (−0.594 + 0.343i)19-s + (−0.0519 + 0.0300i)23-s + (−0.165 + 0.286i)25-s + (0.908 + 0.417i)27-s − 0.382i·29-s + (0.936 + 0.540i)31-s + (−0.0712 + 0.493i)33-s + (−0.144 − 0.249i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.311 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.311 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1357514821\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1357514821\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.642 + 1.60i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (1.28 + 2.23i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1.43 + 0.826i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 5.71iT - 13T^{2} \) |
| 17 | \( 1 + (3.79 - 6.56i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.58 - 1.49i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.249 - 0.143i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 2.05iT - 29T^{2} \) |
| 31 | \( 1 + (-5.21 - 3.00i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.877 + 1.51i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 4.28T + 41T^{2} \) |
| 43 | \( 1 - 2.46T + 43T^{2} \) |
| 47 | \( 1 + (-0.186 - 0.323i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (6.73 + 3.88i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4.89 - 8.48i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (0.889 - 0.513i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.18 - 2.04i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 15.6iT - 71T^{2} \) |
| 73 | \( 1 + (-3.30 - 1.90i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.56 - 7.89i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 6.65T + 83T^{2} \) |
| 89 | \( 1 + (7.25 + 12.5i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 4.43iT - 97T^{2} \) |
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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
−8.788625868668170576434758672040, −8.212262015179503602518657261049, −7.86585054479274509555172577838, −6.61948113966171533350074086833, −5.83362020338801750808931299869, −5.03872438374598167638042224731, −3.99358557960462542009820420802, −2.64947757808049039823215369529, −1.29925755110837620334912486338, −0.06374540205172949860296177821,
2.38072748555930363516942912982, 3.34423775673991057921773341778, 4.43102874630545057234066651606, 4.92950597673553216114658027880, 6.41952539036920101994651445167, 6.80779935906508107350623456673, 7.82230629705767787273246359145, 9.031265205327956209628185684665, 9.465645699877222781651750486246, 10.48056544928321036598026599546