| L(s) = 1 | − 3-s + 5-s + 7-s − 2·9-s − 2·11-s − 13-s − 15-s − 2·17-s − 2·19-s − 21-s + 23-s + 25-s + 5·27-s + 5·29-s + 5·31-s + 2·33-s + 35-s + 8·37-s + 39-s + 5·41-s − 10·43-s − 2·45-s − 47-s + 49-s + 2·51-s − 2·55-s + 2·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s − 2/3·9-s − 0.603·11-s − 0.277·13-s − 0.258·15-s − 0.485·17-s − 0.458·19-s − 0.218·21-s + 0.208·23-s + 1/5·25-s + 0.962·27-s + 0.928·29-s + 0.898·31-s + 0.348·33-s + 0.169·35-s + 1.31·37-s + 0.160·39-s + 0.780·41-s − 1.52·43-s − 0.298·45-s − 0.145·47-s + 1/7·49-s + 0.280·51-s − 0.269·55-s + 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| 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
−7.73166728206970448358735069218, −6.77198789307071441579868544501, −6.23801167469960421592070908855, −5.53557329495927395132232128494, −4.87707606094686534424534353214, −4.27175895263185944283460991661, −2.93330683315518774846338933425, −2.43336744503911114873078243772, −1.21342803724043176879352794281, 0,
1.21342803724043176879352794281, 2.43336744503911114873078243772, 2.93330683315518774846338933425, 4.27175895263185944283460991661, 4.87707606094686534424534353214, 5.53557329495927395132232128494, 6.23801167469960421592070908855, 6.77198789307071441579868544501, 7.73166728206970448358735069218
