L(s) = 1 | + (1.18 + 1.26i)3-s + (−1.68 + 2.92i)5-s + (0.686 + 1.18i)7-s + (−0.186 + 2.99i)9-s + (0.5 + 0.866i)11-s + (2.68 − 4.65i)13-s + (−5.68 + 1.33i)15-s + 0.372·17-s − 6.37·19-s + (−0.686 + 2.27i)21-s + (−2.68 + 4.65i)23-s + (−3.18 − 5.51i)25-s + (−4.00 + 3.31i)27-s + (0.686 + 1.18i)29-s + (−0.313 + 0.543i)31-s + ⋯ |
L(s) = 1 | + (0.684 + 0.728i)3-s + (−0.754 + 1.30i)5-s + (0.259 + 0.449i)7-s + (−0.0620 + 0.998i)9-s + (0.150 + 0.261i)11-s + (0.745 − 1.29i)13-s + (−1.46 + 0.344i)15-s + 0.0902·17-s − 1.46·19-s + (−0.149 + 0.496i)21-s + (−0.560 + 0.970i)23-s + (−0.637 − 1.10i)25-s + (−0.769 + 0.638i)27-s + (0.127 + 0.220i)29-s + (−0.0563 + 0.0976i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.594 - 0.804i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.594 - 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.678353 + 1.34415i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.678353 + 1.34415i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.18 - 1.26i)T \) |
good | 5 | \( 1 + (1.68 - 2.92i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-0.686 - 1.18i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.68 + 4.65i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 0.372T + 17T^{2} \) |
| 19 | \( 1 + 6.37T + 19T^{2} \) |
| 23 | \( 1 + (2.68 - 4.65i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.686 - 1.18i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (0.313 - 0.543i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 2.74T + 37T^{2} \) |
| 41 | \( 1 + (-0.127 + 0.221i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.87 - 8.43i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.686 - 1.18i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 + (-3.5 + 6.06i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.68 + 2.92i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.87 + 6.70i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 - 5.11T + 73T^{2} \) |
| 79 | \( 1 + (-0.313 - 0.543i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-7.68 - 13.3i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + (-4.87 - 8.43i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.81339833182346947573777189244, −10.37538350504438032934305369069, −9.300212503901636284253147105014, −8.209464148200530820515127692093, −7.78779782808171112512966871612, −6.58288246111339101907755245212, −5.46898259604252434834698099105, −4.11006399251522869937958301214, −3.35449600559463227719451255998, −2.34716446829473048557328915308,
0.810704621221195081761531485835, 2.10789157058983036020927388752, 3.96687817422801291273168861651, 4.34377646760899051440474906343, 5.98737232361202366826426560155, 6.95574285787064920799539087091, 7.941038416774272864125033874579, 8.727763246085499249556116870882, 8.980129765778150270013430790979, 10.44072459843414715444952837624