| L(s) = 1 | + (−0.656 − 1.25i)2-s + (1.44 + 0.963i)3-s + (−1.13 + 1.64i)4-s + (1.41 − 1.73i)5-s + (0.259 − 2.43i)6-s + (−2.06 − 0.854i)7-s + (2.80 + 0.345i)8-s + (0.00250 + 0.00605i)9-s + (−3.09 − 0.629i)10-s + (4.25 − 2.84i)11-s + (−3.22 + 1.27i)12-s + (−2.24 + 0.445i)13-s + (0.284 + 3.14i)14-s + (3.70 − 1.13i)15-s + (−1.41 − 3.74i)16-s + (1.49 + 1.49i)17-s + ⋯ |
| L(s) = 1 | + (−0.464 − 0.885i)2-s + (0.832 + 0.556i)3-s + (−0.568 + 0.822i)4-s + (0.631 − 0.775i)5-s + (0.106 − 0.995i)6-s + (−0.779 − 0.322i)7-s + (0.992 + 0.122i)8-s + (0.000835 + 0.00201i)9-s + (−0.979 − 0.199i)10-s + (1.28 − 0.857i)11-s + (−0.930 + 0.368i)12-s + (−0.621 + 0.123i)13-s + (0.0759 + 0.840i)14-s + (0.956 − 0.294i)15-s + (−0.352 − 0.935i)16-s + (0.361 + 0.361i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.341 + 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.341 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.12315 - 0.787099i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.12315 - 0.787099i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.656 + 1.25i)T \) |
| 5 | \( 1 + (-1.41 + 1.73i)T \) |
| good | 3 | \( 1 + (-1.44 - 0.963i)T + (1.14 + 2.77i)T^{2} \) |
| 7 | \( 1 + (2.06 + 0.854i)T + (4.94 + 4.94i)T^{2} \) |
| 11 | \( 1 + (-4.25 + 2.84i)T + (4.20 - 10.1i)T^{2} \) |
| 13 | \( 1 + (2.24 - 0.445i)T + (12.0 - 4.97i)T^{2} \) |
| 17 | \( 1 + (-1.49 - 1.49i)T + 17iT^{2} \) |
| 19 | \( 1 + (-6.78 + 1.34i)T + (17.5 - 7.27i)T^{2} \) |
| 23 | \( 1 + (-1.96 - 4.74i)T + (-16.2 + 16.2i)T^{2} \) |
| 29 | \( 1 + (1.63 + 1.09i)T + (11.0 + 26.7i)T^{2} \) |
| 31 | \( 1 - 4.02iT - 31T^{2} \) |
| 37 | \( 1 + (0.862 - 4.33i)T + (-34.1 - 14.1i)T^{2} \) |
| 41 | \( 1 + (8.18 - 3.39i)T + (28.9 - 28.9i)T^{2} \) |
| 43 | \( 1 + (3.13 - 2.09i)T + (16.4 - 39.7i)T^{2} \) |
| 47 | \( 1 + (4.72 + 4.72i)T + 47iT^{2} \) |
| 53 | \( 1 + (0.185 + 0.278i)T + (-20.2 + 48.9i)T^{2} \) |
| 59 | \( 1 + (0.0216 - 0.108i)T + (-54.5 - 22.5i)T^{2} \) |
| 61 | \( 1 + (4.85 - 7.26i)T + (-23.3 - 56.3i)T^{2} \) |
| 67 | \( 1 + (-4.28 - 2.86i)T + (25.6 + 61.8i)T^{2} \) |
| 71 | \( 1 + (0.720 - 1.73i)T + (-50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (5.06 - 2.09i)T + (51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (-10.3 - 10.3i)T + 79iT^{2} \) |
| 83 | \( 1 + (-2.15 - 10.8i)T + (-76.6 + 31.7i)T^{2} \) |
| 89 | \( 1 + (-14.6 - 6.06i)T + (62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 + 2.02T + 97T^{2} \) |
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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
−11.55868510833694940542131218880, −10.05865975406597529909974764988, −9.580286348927775679783303953726, −9.034569436086668076843619998656, −8.158800392541014861123618797563, −6.71678734367811907152893459462, −5.16464101156995495443087480945, −3.76435648135493061828671474958, −3.09264816064904812750422711053, −1.24857343525174588651616696202,
1.85545896589180617115617125093, 3.26634661286992641481750270699, 5.11738118280861583913292199508, 6.34464997274198512200644396445, 7.08192256187855536678722450622, 7.77168887799742658991010672405, 9.206660193996249610413309683502, 9.493494967065198206510426001277, 10.47658041870659084216053009194, 11.92647311073414077534588715166
