L(s) = 1 | − 3.84·5-s + (−0.676 + 2.55i)7-s + 1.80·11-s + (−0.692 + 1.19i)13-s + (0.833 − 1.44i)17-s + (0.0802 + 0.138i)19-s + 3.20·23-s + 9.75·25-s + (3.78 + 6.54i)29-s + (1.61 + 2.78i)31-s + (2.59 − 9.82i)35-s + (1.58 + 2.74i)37-s + (−6.00 + 10.3i)41-s + (−3.45 − 5.98i)43-s + (−5.71 + 9.90i)47-s + ⋯ |
L(s) = 1 | − 1.71·5-s + (−0.255 + 0.966i)7-s + 0.544·11-s + (−0.192 + 0.332i)13-s + (0.202 − 0.350i)17-s + (0.0184 + 0.0318i)19-s + 0.667·23-s + 1.95·25-s + (0.701 + 1.21i)29-s + (0.289 + 0.500i)31-s + (0.439 − 1.66i)35-s + (0.260 + 0.451i)37-s + (−0.937 + 1.62i)41-s + (−0.526 − 0.912i)43-s + (−0.834 + 1.44i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 + 0.272i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.962 + 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2248875395\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2248875395\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.676 - 2.55i)T \) |
good | 5 | \( 1 + 3.84T + 5T^{2} \) |
| 11 | \( 1 - 1.80T + 11T^{2} \) |
| 13 | \( 1 + (0.692 - 1.19i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.833 + 1.44i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.0802 - 0.138i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 3.20T + 23T^{2} \) |
| 29 | \( 1 + (-3.78 - 6.54i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.61 - 2.78i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.58 - 2.74i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (6.00 - 10.3i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.45 + 5.98i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (5.71 - 9.90i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.37 - 2.38i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (7.53 + 13.0i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.60 + 7.96i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.16 + 10.6i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 6.93T + 71T^{2} \) |
| 73 | \( 1 + (6.22 - 10.7i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-8.03 + 13.9i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (1.45 + 2.51i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (5.04 + 8.73i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.18 - 7.25i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.975914204640792568063474416025, −8.366428444876473106666425753278, −7.74108882734573881700296036165, −6.84752176500317445207668943801, −6.34449455301201861340103828054, −4.97577048002550194448756511538, −4.62468808182130379788377683126, −3.35930860549660982925328506158, −3.03389930262620908854798088556, −1.43208103844999167433360697519,
0.085738505804238146699625907740, 1.10460887081312213889495058921, 2.81493812222992992855935402032, 3.75174080850847650479172746528, 4.14108055357223956268553136984, 4.99606936497273201620825131143, 6.21231871356699321529153292930, 7.06170996804499334038759846580, 7.49933736022890517245023261950, 8.235621590077620660530435504688