| L(s) = 1 | + 0.488·3-s + 5-s − 2.31·7-s − 2.76·9-s + 3.36·11-s − 2.28·13-s + 0.488·15-s − 2.63·17-s − 7.91·19-s − 1.13·21-s + 0.545·23-s + 25-s − 2.81·27-s + 1.82·29-s + 3.87·31-s + 1.64·33-s − 2.31·35-s − 3.59·37-s − 1.11·39-s − 2.40·41-s + 11.5·43-s − 2.76·45-s + 1.45·47-s − 1.64·49-s − 1.28·51-s + 2.92·53-s + 3.36·55-s + ⋯ |
| L(s) = 1 | + 0.282·3-s + 0.447·5-s − 0.874·7-s − 0.920·9-s + 1.01·11-s − 0.633·13-s + 0.126·15-s − 0.638·17-s − 1.81·19-s − 0.246·21-s + 0.113·23-s + 0.200·25-s − 0.541·27-s + 0.339·29-s + 0.696·31-s + 0.285·33-s − 0.391·35-s − 0.590·37-s − 0.178·39-s − 0.375·41-s + 1.76·43-s − 0.411·45-s + 0.212·47-s − 0.235·49-s − 0.180·51-s + 0.401·53-s + 0.453·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.544752299\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.544752299\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 3 | \( 1 - 0.488T + 3T^{2} \) |
| 7 | \( 1 + 2.31T + 7T^{2} \) |
| 11 | \( 1 - 3.36T + 11T^{2} \) |
| 13 | \( 1 + 2.28T + 13T^{2} \) |
| 17 | \( 1 + 2.63T + 17T^{2} \) |
| 19 | \( 1 + 7.91T + 19T^{2} \) |
| 23 | \( 1 - 0.545T + 23T^{2} \) |
| 29 | \( 1 - 1.82T + 29T^{2} \) |
| 31 | \( 1 - 3.87T + 31T^{2} \) |
| 37 | \( 1 + 3.59T + 37T^{2} \) |
| 41 | \( 1 + 2.40T + 41T^{2} \) |
| 43 | \( 1 - 11.5T + 43T^{2} \) |
| 47 | \( 1 - 1.45T + 47T^{2} \) |
| 53 | \( 1 - 2.92T + 53T^{2} \) |
| 59 | \( 1 - 3.70T + 59T^{2} \) |
| 61 | \( 1 + 2.42T + 61T^{2} \) |
| 67 | \( 1 - 2.37T + 67T^{2} \) |
| 71 | \( 1 - 3.66T + 71T^{2} \) |
| 73 | \( 1 - 7.45T + 73T^{2} \) |
| 79 | \( 1 - 5.54T + 79T^{2} \) |
| 83 | \( 1 - 6.05T + 83T^{2} \) |
| 89 | \( 1 - 14.4T + 89T^{2} \) |
| 97 | \( 1 + 1.20T + 97T^{2} \) |
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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
−7.928318628827904478099025188916, −6.92520169998250552068489610090, −6.42046048617736794773740078728, −6.00205631433155219531490626172, −5.01433195999869035969694025235, −4.21212333918738219490591684484, −3.47770040851529926490603265277, −2.56729026704389654093165353445, −2.03752798309100543579629934289, −0.57487627342235064843599530248,
0.57487627342235064843599530248, 2.03752798309100543579629934289, 2.56729026704389654093165353445, 3.47770040851529926490603265277, 4.21212333918738219490591684484, 5.01433195999869035969694025235, 6.00205631433155219531490626172, 6.42046048617736794773740078728, 6.92520169998250552068489610090, 7.928318628827904478099025188916
