L(s) = 1 | + (0.928 + 0.371i)2-s + (−0.759 + 0.876i)3-s + (0.723 + 0.690i)4-s + (−1.03 + 0.531i)6-s + (0.415 + 0.909i)8-s + (−0.0492 − 0.342i)9-s + (−0.738 − 0.380i)11-s + (−1.15 + 0.110i)12-s + (0.0475 + 0.998i)16-s + (0.341 − 0.325i)17-s + (0.0815 − 0.336i)18-s + (−0.0748 − 0.0588i)19-s + (−0.544 − 0.627i)22-s + (−1.11 − 0.326i)24-s + (0.415 − 0.909i)25-s + ⋯ |
L(s) = 1 | + (0.928 + 0.371i)2-s + (−0.759 + 0.876i)3-s + (0.723 + 0.690i)4-s + (−1.03 + 0.531i)6-s + (0.415 + 0.909i)8-s + (−0.0492 − 0.342i)9-s + (−0.738 − 0.380i)11-s + (−1.15 + 0.110i)12-s + (0.0475 + 0.998i)16-s + (0.341 − 0.325i)17-s + (0.0815 − 0.336i)18-s + (−0.0748 − 0.0588i)19-s + (−0.544 − 0.627i)22-s + (−1.11 − 0.326i)24-s + (0.415 − 0.909i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0348 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0348 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.178716867\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.178716867\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.928 - 0.371i)T \) |
| 67 | \( 1 + (-0.841 + 0.540i)T \) |
good | 3 | \( 1 + (0.759 - 0.876i)T + (-0.142 - 0.989i)T^{2} \) |
| 5 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 7 | \( 1 + (-0.235 + 0.971i)T^{2} \) |
| 11 | \( 1 + (0.738 + 0.380i)T + (0.580 + 0.814i)T^{2} \) |
| 13 | \( 1 + (0.327 - 0.945i)T^{2} \) |
| 17 | \( 1 + (-0.341 + 0.325i)T + (0.0475 - 0.998i)T^{2} \) |
| 19 | \( 1 + (0.0748 + 0.0588i)T + (0.235 + 0.971i)T^{2} \) |
| 23 | \( 1 + (0.786 - 0.618i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.327 + 0.945i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (0.0671 + 0.276i)T + (-0.888 + 0.458i)T^{2} \) |
| 43 | \( 1 + (-1.70 - 0.500i)T + (0.841 + 0.540i)T^{2} \) |
| 47 | \( 1 + (-0.928 - 0.371i)T^{2} \) |
| 53 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 59 | \( 1 + (0.827 + 1.81i)T + (-0.654 + 0.755i)T^{2} \) |
| 61 | \( 1 + (-0.580 + 0.814i)T^{2} \) |
| 71 | \( 1 + (-0.0475 - 0.998i)T^{2} \) |
| 73 | \( 1 + (1.28 - 0.663i)T + (0.580 - 0.814i)T^{2} \) |
| 79 | \( 1 + (-0.981 + 0.189i)T^{2} \) |
| 83 | \( 1 + (-0.0800 - 1.68i)T + (-0.995 + 0.0950i)T^{2} \) |
| 89 | \( 1 + (1.21 + 1.40i)T + (-0.142 + 0.989i)T^{2} \) |
| 97 | \( 1 + (0.723 - 1.25i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−11.18733317735588926204891120729, −10.69682907616438465072554114010, −9.758164405754283399666663979696, −8.433762321073903150896051950671, −7.53908752440575771375117946614, −6.38277949641098427665802251669, −5.50422499790549751392157420781, −4.83860026865063916150838642069, −3.87334550023884068552633106899, −2.59319488173354515801319615532,
1.41551709662210133830461713591, 2.80311667849474606731546959897, 4.19261520436264544633650064226, 5.38521575439452501596958529538, 6.00674530016372628018800763409, 7.04070877125344826750435500064, 7.67123798118601420232188653272, 9.228484905577083057917337561726, 10.35893815653419513077585527109, 11.02613502357486987637318271751