L(s) = 1 | + 0.481·5-s + (−2.53 + 0.763i)7-s + 3.38·11-s + (−2.86 + 4.95i)13-s + (−2.75 + 4.77i)17-s + (−2.18 − 3.77i)19-s + 3.62·23-s − 4.76·25-s + (−1.53 − 2.65i)29-s + (−4.67 − 8.09i)31-s + (−1.21 + 0.367i)35-s + (1.48 + 2.57i)37-s + (6.29 − 10.9i)41-s + (−1.90 − 3.30i)43-s + (1.88 − 3.26i)47-s + ⋯ |
L(s) = 1 | + 0.215·5-s + (−0.957 + 0.288i)7-s + 1.01·11-s + (−0.793 + 1.37i)13-s + (−0.668 + 1.15i)17-s + (−0.500 − 0.866i)19-s + 0.756·23-s − 0.953·25-s + (−0.284 − 0.492i)29-s + (−0.839 − 1.45i)31-s + (−0.206 + 0.0621i)35-s + (0.244 + 0.422i)37-s + (0.983 − 1.70i)41-s + (−0.291 − 0.504i)43-s + (0.274 − 0.475i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.783 + 0.620i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.783 + 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2490364358\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2490364358\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.53 - 0.763i)T \) |
good | 5 | \( 1 - 0.481T + 5T^{2} \) |
| 11 | \( 1 - 3.38T + 11T^{2} \) |
| 13 | \( 1 + (2.86 - 4.95i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (2.75 - 4.77i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.18 + 3.77i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 3.62T + 23T^{2} \) |
| 29 | \( 1 + (1.53 + 2.65i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (4.67 + 8.09i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.48 - 2.57i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-6.29 + 10.9i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1.90 + 3.30i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.88 + 3.26i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (5.57 - 9.66i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4.21 + 7.29i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.64 + 6.31i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.28 - 2.22i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 3.94T + 71T^{2} \) |
| 73 | \( 1 + (0.862 - 1.49i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.79 - 4.84i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (0.119 + 0.206i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (0.648 + 1.12i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (7.02 + 12.1i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.692934904149241504386384689010, −7.53292664867747033761832278657, −6.74807066003691686294918886238, −6.32790314142629148050078311232, −5.49521804700775310026405170367, −4.22191636996191113239162056934, −3.91445365652312425446431771591, −2.52023273341085959956108773289, −1.82564729899444136295054638111, −0.07702337051673516838435012330,
1.27374328051858899578293342001, 2.66454401131630348479397670340, 3.34653307011039098456212636240, 4.30157318345812968014129521894, 5.24388426647061864850149798009, 6.03486657500621236986687067989, 6.79818274900007493252529092898, 7.39624571050342819777446089821, 8.250706200778242315532480193801, 9.331657166425241926090472895316