L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + 5-s + (−0.499 + 0.866i)6-s + (0.280 − 0.486i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (−2.06 − 3.57i)11-s + 0.999·12-s + (2.84 − 2.21i)13-s − 0.561·14-s + (−0.5 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (1.56 − 2.70i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + 0.447·5-s + (−0.204 + 0.353i)6-s + (0.106 − 0.183i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.158 − 0.273i)10-s + (−0.621 − 1.07i)11-s + 0.288·12-s + (0.788 − 0.615i)13-s − 0.150·14-s + (−0.129 − 0.223i)15-s + (−0.125 − 0.216i)16-s + (0.378 − 0.655i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.597 + 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.597 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.437310 - 0.870658i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.437310 - 0.870658i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + (-2.84 + 2.21i)T \) |
good | 7 | \( 1 + (-0.280 + 0.486i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.06 + 3.57i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-1.56 + 2.70i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.280 - 0.486i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.34 + 4.05i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.21 + 2.11i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 6.68T + 31T^{2} \) |
| 37 | \( 1 + (-2.06 - 3.57i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (6.12 + 10.6i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.219 - 0.379i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 7T + 47T^{2} \) |
| 53 | \( 1 - 8.56T + 53T^{2} \) |
| 59 | \( 1 + (3.21 - 5.57i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3 - 5.19i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.12 - 1.94i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.56 + 11.3i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 9.36T + 73T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 + 7.12T + 83T^{2} \) |
| 89 | \( 1 + (-9.40 - 16.2i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.56 + 6.16i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.78877789190322311714113654435, −10.45585711313670274008723753872, −9.137227002556938843230327917208, −8.299299386119280621650314120078, −7.42285120217096377999195807512, −6.11221232787929381586546860074, −5.27542527390200184001034118722, −3.64285917120525226684542045420, −2.39061168634491868057978214172, −0.77717718719313302481954487118,
1.85435257115306687980002757043, 3.81911376440310712694617052163, 5.04728888401419224601820863295, 5.85779682163974014792744334163, 6.89118418684099624412996700709, 7.936950627451422516493769393711, 8.991851879325523627060260056120, 9.747369322526125847436991098903, 10.51511819947922643287221899020, 11.42186811371079477005831694229