| L(s) = 1 | + 3-s − 2·4-s − 7-s − 2·9-s + 3·11-s − 2·12-s − 4·13-s + 4·16-s + 2·19-s − 21-s − 23-s − 5·25-s − 5·27-s + 2·28-s − 3·29-s + 5·31-s + 3·33-s + 4·36-s + 2·37-s − 4·39-s − 3·41-s − 43-s − 6·44-s − 6·47-s + 4·48-s − 6·49-s + 8·52-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s − 0.377·7-s − 2/3·9-s + 0.904·11-s − 0.577·12-s − 1.10·13-s + 16-s + 0.458·19-s − 0.218·21-s − 0.208·23-s − 25-s − 0.962·27-s + 0.377·28-s − 0.557·29-s + 0.898·31-s + 0.522·33-s + 2/3·36-s + 0.328·37-s − 0.640·39-s − 0.468·41-s − 0.152·43-s − 0.904·44-s − 0.875·47-s + 0.577·48-s − 6/7·49-s + 1.10·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.197645337\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.197645337\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + T \) |
| 349 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
| 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
−8.066377806301726417213867824275, −7.25785336679178467825433547572, −6.44010836366751953209137112369, −5.64790725297501609356442331516, −5.02754640193234773709504767660, −4.15420655912738727441897210135, −3.56197536411476503818488049628, −2.81529542372135853302032357613, −1.82741071814124920144417236038, −0.51797838061556696611777353618,
0.51797838061556696611777353618, 1.82741071814124920144417236038, 2.81529542372135853302032357613, 3.56197536411476503818488049628, 4.15420655912738727441897210135, 5.02754640193234773709504767660, 5.64790725297501609356442331516, 6.44010836366751953209137112369, 7.25785336679178467825433547572, 8.066377806301726417213867824275
