L(s) = 1 | + (0.675 + 1.17i)5-s − 1.45i·7-s + 3.18i·11-s + (−2.23 − 1.28i)13-s + (−2.08 − 3.61i)17-s + (−2.43 − 3.61i)19-s + (6.49 + 3.75i)23-s + (1.58 − 2.74i)25-s + (0.734 + 0.423i)29-s + 0.351·31-s + (1.70 − 0.985i)35-s − 6.89i·37-s + (4.05 − 2.34i)41-s + (6.52 − 3.76i)43-s + (8.04 + 4.64i)47-s + ⋯ |
L(s) = 1 | + (0.302 + 0.523i)5-s − 0.550i·7-s + 0.961i·11-s + (−0.619 − 0.357i)13-s + (−0.505 − 0.876i)17-s + (−0.559 − 0.828i)19-s + (1.35 + 0.782i)23-s + (0.317 − 0.549i)25-s + (0.136 + 0.0787i)29-s + 0.0632·31-s + (0.288 − 0.166i)35-s − 1.13i·37-s + (0.633 − 0.365i)41-s + (0.995 − 0.574i)43-s + (1.17 + 0.677i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 + 0.523i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.851 + 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.744737984\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.744737984\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (2.43 + 3.61i)T \) |
good | 5 | \( 1 + (-0.675 - 1.17i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + 1.45iT - 7T^{2} \) |
| 11 | \( 1 - 3.18iT - 11T^{2} \) |
| 13 | \( 1 + (2.23 + 1.28i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (2.08 + 3.61i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-6.49 - 3.75i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.734 - 0.423i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 0.351T + 31T^{2} \) |
| 37 | \( 1 + 6.89iT - 37T^{2} \) |
| 41 | \( 1 + (-4.05 + 2.34i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-6.52 + 3.76i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-8.04 - 4.64i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (8.76 + 5.05i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.675 - 1.17i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.29 + 7.43i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.08 - 1.88i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.438 - 0.758i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-6.67 - 11.5i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.52 + 4.37i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 7.22iT - 83T^{2} \) |
| 89 | \( 1 + (-3.81 - 2.20i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (7.79 - 4.49i)T + (48.5 - 84.0i)T^{2} \) |
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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
−8.990901671101890327189090969473, −7.78268271154357125783953575456, −7.06496529348022211309800575818, −6.82730003856956418093572899837, −5.60057114960242113078585467169, −4.81074028851035542818867821473, −4.08547389022192283112794368361, −2.83943541177473253510520974601, −2.22274766033428976934127781580, −0.67054394979848924267737384792,
1.02954827969740152032295753372, 2.19996560152605227194363950967, 3.12686842472775735045789874124, 4.25305653660491137327067730862, 5.00025844027548777079777928654, 5.89265569897071255376477205983, 6.40635876047664326513406535359, 7.42508174328687774146075881015, 8.368311377832680634497105270671, 8.849702835146231007770618421150