L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s − 3.44·5-s − 0.999·8-s + (−1.72 − 2.98i)10-s − 2·11-s + (2.44 + 4.24i)13-s + (−0.5 − 0.866i)16-s + (1 + 1.73i)17-s + (−3.72 + 6.45i)19-s + (1.72 − 2.98i)20-s + (−1 − 1.73i)22-s + 23-s + 6.89·25-s + (−2.44 + 4.24i)26-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s − 1.54·5-s − 0.353·8-s + (−0.545 − 0.944i)10-s − 0.603·11-s + (0.679 + 1.17i)13-s + (−0.125 − 0.216i)16-s + (0.242 + 0.420i)17-s + (−0.854 + 1.48i)19-s + (0.385 − 0.667i)20-s + (−0.213 − 0.369i)22-s + 0.208·23-s + 1.37·25-s + (−0.480 + 0.832i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.130 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.130 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1566360708\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1566360708\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 3.44T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + (-2.44 - 4.24i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.72 - 6.45i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - T + 23T^{2} \) |
| 29 | \( 1 + (-1.44 + 2.51i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (3 - 5.19i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.89 + 6.75i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (4.89 + 8.48i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.44 + 2.51i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (4.89 + 8.48i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.550 + 0.953i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1 - 1.73i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.72 + 9.91i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.55 + 2.68i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 9.89T + 71T^{2} \) |
| 73 | \( 1 + (1.44 + 2.51i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.94 + 6.84i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-1 + 1.73i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (3.55 - 6.14i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.44 + 5.97i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.463106610629964122942584279194, −7.898908002510045402934185142226, −7.22104762358732663527054568475, −6.46741383092908585713269574346, −5.62019106833433941663462205394, −4.60739259439891195816549691231, −3.89472952762302052476656000326, −3.42811137801925666191954089918, −1.87639930386285147495056142462, −0.05383231886163351659597374867,
1.03850406949636762748391715834, 2.80526343340802487095454229029, 3.18743402880307755030931399747, 4.32433601038160921969752134919, 4.78718198808210248617648310748, 5.81545162879195288271768667267, 6.79122354035307239535615442110, 7.69722224195739754434420481050, 8.215134200942930806696543644984, 8.940033723339958849083999901796