| L(s) = 1 | + 1.98·2-s + 1.94·4-s − 1.69·5-s + 1.47·7-s − 0.0998·8-s − 3.37·10-s − 11-s + 1.08·13-s + 2.92·14-s − 4.09·16-s + 6.50·17-s + 1.99·19-s − 3.30·20-s − 1.98·22-s + 4.59·23-s − 2.12·25-s + 2.15·26-s + 2.87·28-s − 5.22·29-s + 5.02·31-s − 7.94·32-s + 12.9·34-s − 2.49·35-s − 0.631·37-s + 3.95·38-s + 0.169·40-s + 4.72·41-s + ⋯ |
| L(s) = 1 | + 1.40·2-s + 0.974·4-s − 0.758·5-s + 0.556·7-s − 0.0353·8-s − 1.06·10-s − 0.301·11-s + 0.300·13-s + 0.782·14-s − 1.02·16-s + 1.57·17-s + 0.456·19-s − 0.739·20-s − 0.423·22-s + 0.957·23-s − 0.424·25-s + 0.422·26-s + 0.542·28-s − 0.969·29-s + 0.901·31-s − 1.40·32-s + 2.21·34-s − 0.422·35-s − 0.103·37-s + 0.641·38-s + 0.0267·40-s + 0.737·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.931993782\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.931993782\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 - 1.98T + 2T^{2} \) |
| 5 | \( 1 + 1.69T + 5T^{2} \) |
| 7 | \( 1 - 1.47T + 7T^{2} \) |
| 13 | \( 1 - 1.08T + 13T^{2} \) |
| 17 | \( 1 - 6.50T + 17T^{2} \) |
| 19 | \( 1 - 1.99T + 19T^{2} \) |
| 23 | \( 1 - 4.59T + 23T^{2} \) |
| 29 | \( 1 + 5.22T + 29T^{2} \) |
| 31 | \( 1 - 5.02T + 31T^{2} \) |
| 37 | \( 1 + 0.631T + 37T^{2} \) |
| 41 | \( 1 - 4.72T + 41T^{2} \) |
| 43 | \( 1 + 0.586T + 43T^{2} \) |
| 47 | \( 1 - 5.79T + 47T^{2} \) |
| 53 | \( 1 + 11.0T + 53T^{2} \) |
| 59 | \( 1 - 13.9T + 59T^{2} \) |
| 67 | \( 1 - 0.913T + 67T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 73 | \( 1 + 4.55T + 73T^{2} \) |
| 79 | \( 1 + 5.39T + 79T^{2} \) |
| 83 | \( 1 - 17.4T + 83T^{2} \) |
| 89 | \( 1 - 8.43T + 89T^{2} \) |
| 97 | \( 1 - 8.31T + 97T^{2} \) |
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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
−7.78141356071298559769432406725, −7.43455054213095514576683104046, −6.43696882812661573479542189654, −5.69236960714412695784091153053, −5.11255758613790209070476244255, −4.50292190942979314256678791471, −3.59119967797337250478462955687, −3.25565631258838039272149787818, −2.15934986930718759623718259624, −0.852803904526786533441048167322,
0.852803904526786533441048167322, 2.15934986930718759623718259624, 3.25565631258838039272149787818, 3.59119967797337250478462955687, 4.50292190942979314256678791471, 5.11255758613790209070476244255, 5.69236960714412695784091153053, 6.43696882812661573479542189654, 7.43455054213095514576683104046, 7.78141356071298559769432406725
