| L(s) = 1 | − 2-s + 4-s − 2·5-s + 7-s − 8-s + 2·10-s − 4·11-s + 6·13-s − 14-s + 16-s + 2·17-s + 4·19-s − 2·20-s + 4·22-s − 25-s − 6·26-s + 28-s + 2·29-s − 32-s − 2·34-s − 2·35-s + 10·37-s − 4·38-s + 2·40-s + 6·41-s + 4·43-s − 4·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.377·7-s − 0.353·8-s + 0.632·10-s − 1.20·11-s + 1.66·13-s − 0.267·14-s + 1/4·16-s + 0.485·17-s + 0.917·19-s − 0.447·20-s + 0.852·22-s − 1/5·25-s − 1.17·26-s + 0.188·28-s + 0.371·29-s − 0.176·32-s − 0.342·34-s − 0.338·35-s + 1.64·37-s − 0.648·38-s + 0.316·40-s + 0.937·41-s + 0.609·43-s − 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 66654 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66654 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.673578828\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.673578828\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
| 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
−14.23406642326220, −13.61125648464874, −13.24564525505032, −12.58956629910517, −12.07401862860343, −11.46543807812430, −11.20831274001832, −10.65829519983033, −10.27075156799812, −9.520279354827383, −9.085484718384670, −8.334948220609833, −8.091883020867864, −7.576193767933165, −7.274016504748076, −6.321682517917712, −5.845670714770711, −5.375891361697304, −4.482613952765056, −4.030064063318675, −3.229114745488114, −2.835250679983705, −1.937248156687447, −1.074500940917027, −0.5824027663472188,
0.5824027663472188, 1.074500940917027, 1.937248156687447, 2.835250679983705, 3.229114745488114, 4.030064063318675, 4.482613952765056, 5.375891361697304, 5.845670714770711, 6.321682517917712, 7.274016504748076, 7.576193767933165, 8.091883020867864, 8.334948220609833, 9.085484718384670, 9.520279354827383, 10.27075156799812, 10.65829519983033, 11.20831274001832, 11.46543807812430, 12.07401862860343, 12.58956629910517, 13.24564525505032, 13.61125648464874, 14.23406642326220
