| L(s) = 1 | + (−0.444 − 1.34i)2-s + (−1.60 + 1.19i)4-s − i·5-s − 1.61·7-s + (2.31 + 1.62i)8-s + (−1.34 + 0.444i)10-s + 2.72i·11-s − 6.12i·13-s + (0.718 + 2.16i)14-s + (1.15 − 3.83i)16-s + 4.31·17-s − 2.87i·19-s + (1.19 + 1.60i)20-s + (3.65 − 1.21i)22-s − 7.44·23-s + ⋯ |
| L(s) = 1 | + (−0.314 − 0.949i)2-s + (−0.802 + 0.596i)4-s − 0.447i·5-s − 0.610·7-s + (0.818 + 0.574i)8-s + (−0.424 + 0.140i)10-s + 0.821i·11-s − 1.70i·13-s + (0.192 + 0.579i)14-s + (0.287 − 0.957i)16-s + 1.04·17-s − 0.660i·19-s + (0.266 + 0.358i)20-s + (0.779 − 0.258i)22-s − 1.55·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.818 - 0.574i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.818 - 0.574i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3911235412\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3911235412\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.444 + 1.34i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + iT \) |
| good | 7 | \( 1 + 1.61T + 7T^{2} \) |
| 11 | \( 1 - 2.72iT - 11T^{2} \) |
| 13 | \( 1 + 6.12iT - 13T^{2} \) |
| 17 | \( 1 - 4.31T + 17T^{2} \) |
| 19 | \( 1 + 2.87iT - 19T^{2} \) |
| 23 | \( 1 + 7.44T + 23T^{2} \) |
| 29 | \( 1 - 1.43iT - 29T^{2} \) |
| 31 | \( 1 + 9.17T + 31T^{2} \) |
| 37 | \( 1 - 9.26iT - 37T^{2} \) |
| 41 | \( 1 + 6.77T + 41T^{2} \) |
| 43 | \( 1 + 7.56iT - 43T^{2} \) |
| 47 | \( 1 - 0.642T + 47T^{2} \) |
| 53 | \( 1 + 10.0iT - 53T^{2} \) |
| 59 | \( 1 - 14.5iT - 59T^{2} \) |
| 61 | \( 1 + 9.00iT - 61T^{2} \) |
| 67 | \( 1 + 6.55iT - 67T^{2} \) |
| 71 | \( 1 + 3.13T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 + 1.53T + 79T^{2} \) |
| 83 | \( 1 - 8.44iT - 83T^{2} \) |
| 89 | \( 1 + 7.56T + 89T^{2} \) |
| 97 | \( 1 + 1.11T + 97T^{2} \) |
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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
−9.792493892535220810771750784661, −8.633951108447839623732672539196, −7.966854291320270699190547161044, −7.13377998997142832236998401929, −5.69547000911964260076473357979, −4.96228687242651163776608950814, −3.76190445886785016012604920957, −2.98071514499721206978542227610, −1.67375539712374949087347265798, −0.19355587932377680281059335118,
1.73695221072778335997116732780, 3.48477221838247008481393798529, 4.25456500306481087354933431126, 5.69183235591328202354592754789, 6.13018726934411756621309470783, 7.05268501787687500059807592110, 7.77042101935244104642535785252, 8.695218368142716013001372621223, 9.490453250646842247187522832454, 10.05483211173457612890477598022
