| L(s) = 1 | + (0.996 + 0.0871i)2-s + (−1.08 − 1.35i)3-s + (0.984 + 0.173i)4-s + (−2.15 − 0.611i)5-s + (−0.963 − 1.43i)6-s + (−3.60 − 2.52i)7-s + (0.965 + 0.258i)8-s + (−0.646 + 2.92i)9-s + (−2.08 − 0.796i)10-s + (0.474 − 1.30i)11-s + (−0.833 − 1.51i)12-s + (−0.410 − 4.69i)13-s + (−3.36 − 2.82i)14-s + (1.50 + 3.56i)15-s + (0.939 + 0.342i)16-s + (5.62 − 1.50i)17-s + ⋯ |
| L(s) = 1 | + (0.704 + 0.0616i)2-s + (−0.626 − 0.779i)3-s + (0.492 + 0.0868i)4-s + (−0.961 − 0.273i)5-s + (−0.393 − 0.587i)6-s + (−1.36 − 0.953i)7-s + (0.341 + 0.0915i)8-s + (−0.215 + 0.976i)9-s + (−0.660 − 0.251i)10-s + (0.143 − 0.393i)11-s + (−0.240 − 0.438i)12-s + (−0.113 − 1.30i)13-s + (−0.900 − 0.755i)14-s + (0.389 + 0.921i)15-s + (0.234 + 0.0855i)16-s + (1.36 − 0.365i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.680 + 0.732i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.680 + 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.363306 - 0.833684i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.363306 - 0.833684i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.996 - 0.0871i)T \) |
| 3 | \( 1 + (1.08 + 1.35i)T \) |
| 5 | \( 1 + (2.15 + 0.611i)T \) |
| good | 7 | \( 1 + (3.60 + 2.52i)T + (2.39 + 6.57i)T^{2} \) |
| 11 | \( 1 + (-0.474 + 1.30i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (0.410 + 4.69i)T + (-12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (-5.62 + 1.50i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (7.17 - 4.14i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.511 - 0.730i)T + (-7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (-4.04 + 3.39i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-1.07 + 6.11i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-0.319 - 1.19i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-1.37 + 1.63i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (1.59 + 3.41i)T + (-27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (-2.26 + 3.23i)T + (-16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (6.56 - 6.56i)T - 53iT^{2} \) |
| 59 | \( 1 + (-2.97 + 1.08i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (0.104 + 0.590i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (1.40 - 0.122i)T + (65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (-10.7 - 6.20i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.128 + 0.481i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (7.34 + 8.75i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-0.841 + 9.61i)T + (-81.7 - 14.4i)T^{2} \) |
| 89 | \( 1 + (-3.02 - 5.24i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (3.52 - 1.64i)T + (62.3 - 74.3i)T^{2} \) |
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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
−11.92312675806580038826100473802, −10.76666078120617935762184161709, −10.08202114821796053046259995496, −8.176782999389268547481426287593, −7.51933591942741353971890489406, −6.48731179652568632426788034547, −5.63145655185429090254083192167, −4.15843880630540749803628540220, −3.10085911677825735997097379896, −0.59429507533099108736138336568,
2.91620345588969918315596789349, 3.95592058770276953320880262225, 4.95498370890109934075471069652, 6.35309978308886628961519990178, 6.80283905413211652182971751414, 8.613139990146547106608500002878, 9.573433838259141062565574080070, 10.56506632435701292739516634633, 11.49173578541459841350980845987, 12.36719710458014738829163640996
