| L(s) = 1 | + (−0.104 − 0.994i)2-s + (0.110 − 1.72i)3-s + (−0.978 + 0.207i)4-s + (−1.75 − 3.94i)5-s + (−1.73 + 0.0705i)6-s + (2.64 − 0.0650i)7-s + (0.309 + 0.951i)8-s + (−2.97 − 0.382i)9-s + (−3.73 + 2.15i)10-s + (3.29 − 0.387i)11-s + (0.251 + 1.71i)12-s + (−2.24 − 3.09i)13-s + (−0.341 − 2.62i)14-s + (−7.01 + 2.59i)15-s + (0.913 − 0.406i)16-s + (−0.195 + 1.86i)17-s + ⋯ |
| L(s) = 1 | + (−0.0739 − 0.703i)2-s + (0.0639 − 0.997i)3-s + (−0.489 + 0.103i)4-s + (−0.785 − 1.76i)5-s + (−0.706 + 0.0288i)6-s + (0.999 − 0.0245i)7-s + (0.109 + 0.336i)8-s + (−0.991 − 0.127i)9-s + (−1.18 + 0.682i)10-s + (0.993 − 0.116i)11-s + (0.0724 + 0.494i)12-s + (−0.623 − 0.858i)13-s + (−0.0911 − 0.701i)14-s + (−1.81 + 0.671i)15-s + (0.228 − 0.101i)16-s + (−0.0474 + 0.451i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 - 0.308i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.951 - 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.178625 + 1.13169i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.178625 + 1.13169i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.104 + 0.994i)T \) |
| 3 | \( 1 + (-0.110 + 1.72i)T \) |
| 7 | \( 1 + (-2.64 + 0.0650i)T \) |
| 11 | \( 1 + (-3.29 + 0.387i)T \) |
| good | 5 | \( 1 + (1.75 + 3.94i)T + (-3.34 + 3.71i)T^{2} \) |
| 13 | \( 1 + (2.24 + 3.09i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (0.195 - 1.86i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (-0.385 + 1.81i)T + (-17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (-5.76 - 3.33i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.60 - 4.92i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-1.16 - 0.519i)T + (20.7 + 23.0i)T^{2} \) |
| 37 | \( 1 + (-3.11 - 3.45i)T + (-3.86 + 36.7i)T^{2} \) |
| 41 | \( 1 + (2.90 + 8.94i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 2.90iT - 43T^{2} \) |
| 47 | \( 1 + (1.28 - 6.05i)T + (-42.9 - 19.1i)T^{2} \) |
| 53 | \( 1 + (-5.16 + 11.5i)T + (-35.4 - 39.3i)T^{2} \) |
| 59 | \( 1 + (-0.0588 - 0.276i)T + (-53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (3.90 + 8.77i)T + (-40.8 + 45.3i)T^{2} \) |
| 67 | \( 1 + (3.82 + 6.61i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.878 + 1.20i)T + (-21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-1.25 - 5.92i)T + (-66.6 + 29.6i)T^{2} \) |
| 79 | \( 1 + (-4.12 + 0.433i)T + (77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (4.25 + 3.08i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (-0.326 - 0.188i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (3.32 - 2.41i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.96943010026056941604351203460, −9.340329424751673499326828668001, −8.681361148540975120198214026435, −8.063891266211788252879575790438, −7.20236873234385723633188634633, −5.44887605962724352298950121926, −4.78801077539104363428364317572, −3.51153923162082092697294535942, −1.69245835232227612409657887569, −0.78708630729016095726032503755,
2.67664123645642413845522657614, 3.97873583083535414264211828553, 4.63947887480686838367367987350, 6.09112831678068214360689327621, 7.01850609249468000299971445891, 7.76751339765527867447204108187, 8.815548462273600158488816200335, 9.764693122860352602022261579975, 10.64063406886673713413258378589, 11.47481560599496197123233960815
