L(s) = 1 | + (−0.927 + 2.85i)3-s + (−0.618 − 1.90i)4-s + (0.809 − 0.587i)5-s + (−0.309 − 0.951i)7-s + (−4.85 − 3.52i)9-s + 6·12-s + (3.23 + 2.35i)13-s + (0.927 + 2.85i)15-s + (−3.23 + 2.35i)16-s + (−1.61 + 1.17i)17-s + (−1.85 + 5.70i)19-s + (−1.61 − 1.17i)20-s + 3·21-s − 5·23-s + (−1.23 + 3.80i)25-s + ⋯ |
L(s) = 1 | + (−0.535 + 1.64i)3-s + (−0.309 − 0.951i)4-s + (0.361 − 0.262i)5-s + (−0.116 − 0.359i)7-s + (−1.61 − 1.17i)9-s + 1.73·12-s + (0.897 + 0.652i)13-s + (0.239 + 0.736i)15-s + (−0.809 + 0.587i)16-s + (−0.392 + 0.285i)17-s + (−0.425 + 1.30i)19-s + (−0.361 − 0.262i)20-s + 0.654·21-s − 1.04·23-s + (−0.247 + 0.760i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.859 - 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.185883 + 0.675850i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.185883 + 0.675850i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (0.309 + 0.951i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.618 + 1.90i)T^{2} \) |
| 3 | \( 1 + (0.927 - 2.85i)T + (-2.42 - 1.76i)T^{2} \) |
| 5 | \( 1 + (-0.809 + 0.587i)T + (1.54 - 4.75i)T^{2} \) |
| 13 | \( 1 + (-3.23 - 2.35i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (1.61 - 1.17i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (1.85 - 5.70i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + 5T + 23T^{2} \) |
| 29 | \( 1 + (-3.09 - 9.51i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (0.809 + 0.587i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (1.54 + 4.75i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (0.618 - 1.90i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + (-2.47 + 7.60i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-4.85 - 3.52i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-0.927 - 2.85i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-1.61 + 1.17i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + 3T + 67T^{2} \) |
| 71 | \( 1 + (0.809 - 0.587i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-3.09 - 9.51i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (4.85 + 3.52i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (9.70 - 7.05i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + 15T + 89T^{2} \) |
| 97 | \( 1 + (-4.04 - 2.93i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.35390066447151499866308717595, −9.953118478650024804061516136079, −9.057197029011564538102728639270, −8.524732385927276877734063511625, −6.71843770074205831825508880950, −5.83414094097184979341264991970, −5.29667401117005760025988128442, −4.24545614410316390284225995878, −3.69280650178484932255531239105, −1.59698148138712819478518166525,
0.36429114385404090000695432094, 2.11043495212053107590904379756, 2.95038797261019570639302863580, 4.47588821917921845466291307550, 5.80029400882931160556507219633, 6.45746843615157233125644712578, 7.18583560045705690708390309795, 8.202350376194745578808427725414, 8.508697939236556891074884935004, 9.804263758573919835765140791742