L(s) = 1 | + (0.105 − 0.0611i)2-s + (−0.992 + 1.71i)4-s − 0.529·5-s + 0.487i·8-s + (−0.0560 + 0.0323i)10-s − 4.20i·11-s + (−1.74 + 1.00i)13-s + (−1.95 − 3.38i)16-s + (2.19 + 3.79i)17-s + (−4.54 − 2.62i)19-s + (0.525 − 0.910i)20-s + (−0.257 − 0.445i)22-s − 6.27i·23-s − 4.71·25-s + (−0.123 + 0.213i)26-s + ⋯ |
L(s) = 1 | + (0.0749 − 0.0432i)2-s + (−0.496 + 0.859i)4-s − 0.236·5-s + 0.172i·8-s + (−0.0177 + 0.0102i)10-s − 1.26i·11-s + (−0.484 + 0.279i)13-s + (−0.488 − 0.846i)16-s + (0.532 + 0.921i)17-s + (−1.04 − 0.601i)19-s + (0.117 − 0.203i)20-s + (−0.0548 − 0.0949i)22-s − 1.30i·23-s − 0.943·25-s + (−0.0242 + 0.0419i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0635 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0635 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7961456924\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7961456924\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.105 + 0.0611i)T + (1 - 1.73i)T^{2} \) |
| 5 | \( 1 + 0.529T + 5T^{2} \) |
| 11 | \( 1 + 4.20iT - 11T^{2} \) |
| 13 | \( 1 + (1.74 - 1.00i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.19 - 3.79i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (4.54 + 2.62i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 6.27iT - 23T^{2} \) |
| 29 | \( 1 + (-7.27 - 4.20i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1.03 + 0.595i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.61 + 2.79i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.0994 - 0.172i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.96 + 6.86i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (4.98 + 8.63i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.65 + 2.10i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.71 + 11.6i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (11.3 - 6.55i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.29 + 5.69i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 8.50iT - 71T^{2} \) |
| 73 | \( 1 + (-4.86 + 2.80i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.286 + 0.495i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (5.42 - 9.39i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-6.43 + 11.1i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.493 - 0.285i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.210128179145594379524645088291, −8.452824505020598274127621848377, −8.131806320857075135372391715151, −7.00788678884004330459268997282, −6.18213916834387926597512268751, −5.06838571237407014151559524275, −4.15117500823409137576484766971, −3.40015494951497328731226364455, −2.32389259975456805028065747490, −0.34180931942456184879886322920,
1.32622610427092126020511235831, 2.57719076106772562993372270982, 4.06056631889958115962458750695, 4.71639657006925010367978503402, 5.58885741684692672187543611301, 6.44720283762218010694665859492, 7.44668001219216109375753561590, 8.111967509600669927954093750366, 9.298768887068039703433624092060, 9.822374304080883989527009730187