L(s) = 1 | + (−0.5 + 0.133i)2-s + (−1.5 + 0.866i)4-s + (−0.133 + 2.23i)5-s + (−2.5 − 0.866i)7-s + (1.36 − 1.36i)8-s + (−0.232 − 1.13i)10-s + (−1.36 − 2.36i)11-s + (−2 − 2i)13-s + (1.36 + 0.0980i)14-s + (1.23 − 2.13i)16-s + (−3.73 − i)17-s + (0.366 − 0.633i)19-s + (−1.73 − 3.46i)20-s + (1 + i)22-s + (−0.0358 − 0.133i)23-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.0947i)2-s + (−0.750 + 0.433i)4-s + (−0.0599 + 0.998i)5-s + (−0.944 − 0.327i)7-s + (0.482 − 0.482i)8-s + (−0.0733 − 0.358i)10-s + (−0.411 − 0.713i)11-s + (−0.554 − 0.554i)13-s + (0.365 + 0.0262i)14-s + (0.308 − 0.533i)16-s + (−0.905 − 0.242i)17-s + (0.0839 − 0.145i)19-s + (−0.387 − 0.774i)20-s + (0.213 + 0.213i)22-s + (−0.00748 − 0.0279i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.848 + 0.528i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.848 + 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.133 - 2.23i)T \) |
| 7 | \( 1 + (2.5 + 0.866i)T \) |
good | 2 | \( 1 + (0.5 - 0.133i)T + (1.73 - i)T^{2} \) |
| 11 | \( 1 + (1.36 + 2.36i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2 + 2i)T + 13iT^{2} \) |
| 17 | \( 1 + (3.73 + i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-0.366 + 0.633i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.0358 + 0.133i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 - 3iT - 29T^{2} \) |
| 31 | \( 1 + (6.46 - 3.73i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4.73 - 1.26i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 - 6.46iT - 41T^{2} \) |
| 43 | \( 1 + (-2.83 + 2.83i)T - 43iT^{2} \) |
| 47 | \( 1 + (2.36 + 8.83i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (6.83 + 1.83i)T + (45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (-4.09 - 7.09i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.33 - 0.767i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.86 + 10.6i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + 1.26T + 71T^{2} \) |
| 73 | \( 1 + (3.46 - 12.9i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-2.83 - 1.63i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.09 - 2.09i)T + 83iT^{2} \) |
| 89 | \( 1 + (-0.330 + 0.571i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (5.92 - 5.92i)T - 97iT^{2} \) |
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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
−11.01988965915660683459773698953, −10.28300927063859411199442970330, −9.452678417596889087818756325419, −8.458790976017573506282010299547, −7.38445168254937815534997134423, −6.66949696420315285969735052589, −5.27088811747544515284029233239, −3.77232174949928091027752476008, −2.89786278698498754613310790277, 0,
2.01241289316343666063431252314, 4.05757220432945931809831945093, 4.98297382150262257174462376505, 6.01342747302789142007375845135, 7.40934533161736134437569534928, 8.561105879653770955758932865974, 9.372860037563794455311362211296, 9.792981239999169455567177328440, 10.97889214881415987757610845828