| L(s) = 1 | − 2·3-s + 2·7-s + 9-s + 11-s + 4·17-s − 4·19-s − 4·21-s − 6·23-s + 4·27-s + 2·29-s − 8·31-s − 2·33-s + 4·37-s − 6·41-s − 6·43-s − 2·47-s − 3·49-s − 8·51-s + 12·53-s + 8·57-s − 4·59-s + 14·61-s + 2·63-s + 10·67-s + 12·69-s − 8·71-s − 4·73-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.755·7-s + 1/3·9-s + 0.301·11-s + 0.970·17-s − 0.917·19-s − 0.872·21-s − 1.25·23-s + 0.769·27-s + 0.371·29-s − 1.43·31-s − 0.348·33-s + 0.657·37-s − 0.937·41-s − 0.914·43-s − 0.291·47-s − 3/7·49-s − 1.12·51-s + 1.64·53-s + 1.05·57-s − 0.520·59-s + 1.79·61-s + 0.251·63-s + 1.22·67-s + 1.44·69-s − 0.949·71-s − 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
| 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
−8.113713101478193504843364183198, −7.11434235841805371345044756514, −6.45935189086764972305495004639, −5.67476624642674356324852326751, −5.21889555571648073776472930146, −4.35457823872727223501946287577, −3.56064258030205737889582458886, −2.23586903774826597459581614384, −1.26072011469527840778407418232, 0,
1.26072011469527840778407418232, 2.23586903774826597459581614384, 3.56064258030205737889582458886, 4.35457823872727223501946287577, 5.21889555571648073776472930146, 5.67476624642674356324852326751, 6.45935189086764972305495004639, 7.11434235841805371345044756514, 8.113713101478193504843364183198
