| L(s) = 1 | − 2-s + 4-s − 5-s + 4·7-s − 8-s + 10-s + 4·11-s + 13-s − 4·14-s + 16-s + 17-s + 8·19-s − 20-s − 4·22-s + 4·23-s + 25-s − 26-s + 4·28-s − 2·29-s − 32-s − 34-s − 4·35-s + 6·37-s − 8·38-s + 40-s + 6·41-s + 4·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.51·7-s − 0.353·8-s + 0.316·10-s + 1.20·11-s + 0.277·13-s − 1.06·14-s + 1/4·16-s + 0.242·17-s + 1.83·19-s − 0.223·20-s − 0.852·22-s + 0.834·23-s + 1/5·25-s − 0.196·26-s + 0.755·28-s − 0.371·29-s − 0.176·32-s − 0.171·34-s − 0.676·35-s + 0.986·37-s − 1.29·38-s + 0.158·40-s + 0.937·41-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.651091459\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.651091459\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−15.72902299939132, −15.11617953766214, −14.58520347528034, −14.22352481758653, −13.68743607752280, −12.76339255078786, −12.18218535346274, −11.53848791459575, −11.27450678534293, −10.97035173132025, −9.960909422345799, −9.462429490746708, −8.943141334223526, −8.273479272576833, −7.856435485333083, −7.223345317650988, −6.811948094087745, −5.768359664734476, −5.334938601533725, −4.484008974470994, −3.884799153827861, −3.090848710786743, −2.186566037414344, −1.171483320374102, −0.9723925203847999,
0.9723925203847999, 1.171483320374102, 2.186566037414344, 3.090848710786743, 3.884799153827861, 4.484008974470994, 5.334938601533725, 5.768359664734476, 6.811948094087745, 7.223345317650988, 7.856435485333083, 8.273479272576833, 8.943141334223526, 9.462429490746708, 9.960909422345799, 10.97035173132025, 11.27450678534293, 11.53848791459575, 12.18218535346274, 12.76339255078786, 13.68743607752280, 14.22352481758653, 14.58520347528034, 15.11617953766214, 15.72902299939132
