| L(s) = 1 | − 2-s + 3-s + 4-s + 3·5-s − 6-s − 7-s − 8-s − 2·9-s − 3·10-s − 2·11-s + 12-s + 14-s + 3·15-s + 16-s + 3·17-s + 2·18-s + 3·20-s − 21-s + 2·22-s − 2·23-s − 24-s + 4·25-s − 5·27-s − 28-s + 9·29-s − 3·30-s − 5·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.34·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s − 2/3·9-s − 0.948·10-s − 0.603·11-s + 0.288·12-s + 0.267·14-s + 0.774·15-s + 1/4·16-s + 0.727·17-s + 0.471·18-s + 0.670·20-s − 0.218·21-s + 0.426·22-s − 0.417·23-s − 0.204·24-s + 4/5·25-s − 0.962·27-s − 0.188·28-s + 1.67·29-s − 0.547·30-s − 0.898·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207214 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207214 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.927446432\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.927446432\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.08408006402804, −12.66533916668934, −12.04790635416700, −11.61726061152263, −10.97945309909894, −10.51433594708505, −10.07344824847673, −9.691620859400326, −9.365116497377468, −8.801907965051553, −8.359680712308944, −7.921761211366732, −7.465866615075627, −6.690422429510577, −6.383221112259715, −5.781706087155588, −5.410720925496728, −4.952403307765779, −3.936606840174733, −3.473259158625693, −2.696680382427439, −2.439254391423094, −2.006777870755986, −1.099642364762640, −0.5532862635731071,
0.5532862635731071, 1.099642364762640, 2.006777870755986, 2.439254391423094, 2.696680382427439, 3.473259158625693, 3.936606840174733, 4.952403307765779, 5.410720925496728, 5.781706087155588, 6.383221112259715, 6.690422429510577, 7.465866615075627, 7.921761211366732, 8.359680712308944, 8.801907965051553, 9.365116497377468, 9.691620859400326, 10.07344824847673, 10.51433594708505, 10.97945309909894, 11.61726061152263, 12.04790635416700, 12.66533916668934, 13.08408006402804
