| L(s) = 1 | + 2-s + 4-s + 8-s − 4·11-s + 2·13-s + 16-s + 17-s + 4·19-s − 4·22-s + 2·26-s + 2·29-s + 8·31-s + 32-s + 34-s − 6·37-s + 4·38-s + 6·41-s + 4·43-s − 4·44-s − 7·49-s + 2·52-s − 10·53-s + 2·58-s + 4·59-s − 2·61-s + 8·62-s + 64-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 1.20·11-s + 0.554·13-s + 1/4·16-s + 0.242·17-s + 0.917·19-s − 0.852·22-s + 0.392·26-s + 0.371·29-s + 1.43·31-s + 0.176·32-s + 0.171·34-s − 0.986·37-s + 0.648·38-s + 0.937·41-s + 0.609·43-s − 0.603·44-s − 49-s + 0.277·52-s − 1.37·53-s + 0.262·58-s + 0.520·59-s − 0.256·61-s + 1.01·62-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.247798499\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.247798499\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 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 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 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
−7.83913131505049522715757122171, −7.15825996977061047116742541437, −6.32223154011246265556733738897, −5.74471986099900548238940946283, −5.00343227083387888486249467899, −4.47235718915283566193943382568, −3.39362999799049909610760132956, −2.92266745018875025733926796747, −1.95779475977991868480725452799, −0.811769753893461137231110538061,
0.811769753893461137231110538061, 1.95779475977991868480725452799, 2.92266745018875025733926796747, 3.39362999799049909610760132956, 4.47235718915283566193943382568, 5.00343227083387888486249467899, 5.74471986099900548238940946283, 6.32223154011246265556733738897, 7.15825996977061047116742541437, 7.83913131505049522715757122171
