L(s) = 1 | + 2-s − 1.73i·3-s + 4-s + (−1.5 − 2.59i)5-s − 1.73i·6-s + (0.5 + 0.866i)7-s + 8-s − 2.99·9-s + (−1.5 − 2.59i)10-s + (−1.5 − 2.59i)11-s − 1.73i·12-s + 2·13-s + (0.5 + 0.866i)14-s + (−4.5 + 2.59i)15-s + 16-s + (−1.5 + 2.59i)17-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.999i·3-s + 0.5·4-s + (−0.670 − 1.16i)5-s − 0.707i·6-s + (0.188 + 0.327i)7-s + 0.353·8-s − 0.999·9-s + (−0.474 − 0.821i)10-s + (−0.452 − 0.783i)11-s − 0.499i·12-s + 0.554·13-s + (0.133 + 0.231i)14-s + (−1.16 + 0.670i)15-s + 0.250·16-s + (−0.363 + 0.630i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.248 + 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.248 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.05958 - 1.36579i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.05958 - 1.36579i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + 1.73iT \) |
| 19 | \( 1 + (-4 + 1.73i)T \) |
good | 5 | \( 1 + (1.5 + 2.59i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-0.5 - 0.866i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + (-4.5 + 7.79i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + (-4.5 - 7.79i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + (-1.5 + 2.59i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.5 - 7.79i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1.5 + 2.59i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5.5 - 9.52i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + (-1.5 + 2.59i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (5.5 - 9.52i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + (7.5 + 12.9i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-4.5 - 7.79i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 14T + 97T^{2} \) |
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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
−11.69477334191851765184303715598, −10.76145791144681951526794130082, −9.001395434818716500242516172941, −8.289259653787766499097660749067, −7.53987392766137708330114210399, −6.19217378527332211864082435408, −5.40835175498338915221951892757, −4.21244237190805062314036927062, −2.76543899002259454848492904008, −1.05019635332378987582131711511,
2.72177976337558570604061299460, 3.67681377935908896639281922401, 4.61416119866816899596961215728, 5.74563138312029592233746261665, 7.03301929568367147840238042383, 7.75329105504494394023281641909, 9.175102276461336140923459036792, 10.31880210177407533118132927649, 10.90022963894186569855411747079, 11.56356557890542592492963326613