| L(s) = 1 | + 2-s − 4-s + 5-s − 7-s − 3·8-s + 10-s + 4·11-s + 2·13-s − 14-s − 16-s + 8·19-s − 20-s + 4·22-s − 4·23-s + 25-s + 2·26-s + 28-s + 2·29-s + 8·31-s + 5·32-s − 35-s − 2·37-s + 8·38-s − 3·40-s − 6·41-s + 4·43-s − 4·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.447·5-s − 0.377·7-s − 1.06·8-s + 0.316·10-s + 1.20·11-s + 0.554·13-s − 0.267·14-s − 1/4·16-s + 1.83·19-s − 0.223·20-s + 0.852·22-s − 0.834·23-s + 1/5·25-s + 0.392·26-s + 0.188·28-s + 0.371·29-s + 1.43·31-s + 0.883·32-s − 0.169·35-s − 0.328·37-s + 1.29·38-s − 0.474·40-s − 0.937·41-s + 0.609·43-s − 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.02316451457891, −13.66982584903606, −13.37764534768752, −12.54141291710861, −12.26423456494442, −11.72841068831257, −11.44614865997075, −10.57549219777333, −10.00128022347898, −9.537211822354981, −9.305173563663221, −8.605464480568825, −8.210109248319277, −7.464617872167720, −6.799103111469363, −6.214070230283035, −5.997384381261360, −5.323188039219843, −4.717661405372286, −4.266184120000235, −3.519436900237673, −3.232036378822654, −2.564136536281183, −1.468372085807374, −1.070515710811789, 0,
1.070515710811789, 1.468372085807374, 2.564136536281183, 3.232036378822654, 3.519436900237673, 4.266184120000235, 4.717661405372286, 5.323188039219843, 5.997384381261360, 6.214070230283035, 6.799103111469363, 7.464617872167720, 8.210109248319277, 8.605464480568825, 9.305173563663221, 9.537211822354981, 10.00128022347898, 10.57549219777333, 11.44614865997075, 11.72841068831257, 12.26423456494442, 12.54141291710861, 13.37764534768752, 13.66982584903606, 14.02316451457891
