L(s) = 1 | − 3-s + 5-s + 7-s + 9-s + 2·13-s − 15-s − 2·17-s + 4·19-s − 21-s + 25-s − 27-s − 6·29-s + 35-s + 6·37-s − 2·39-s + 6·41-s − 4·43-s + 45-s + 49-s + 2·51-s − 2·53-s − 4·57-s − 4·59-s − 6·61-s + 63-s + 2·65-s − 12·67-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.554·13-s − 0.258·15-s − 0.485·17-s + 0.917·19-s − 0.218·21-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.169·35-s + 0.986·37-s − 0.320·39-s + 0.937·41-s − 0.609·43-s + 0.149·45-s + 1/7·49-s + 0.280·51-s − 0.274·53-s − 0.529·57-s − 0.520·59-s − 0.768·61-s + 0.125·63-s + 0.248·65-s − 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 203280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 203280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 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 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−13.26774098859784, −12.94808210357142, −12.22491645624514, −11.87541927437911, −11.33486807547579, −11.00283126662641, −10.57721095412866, −10.02902342488597, −9.481593602191633, −9.119627065197235, −8.657605521512483, −7.899808375852752, −7.563336531423349, −7.084967542341621, −6.407742440826966, −5.910991540161546, −5.722802635993704, −4.890164402488925, −4.648106469905472, −3.946244052695037, −3.361386741673103, −2.731886748347057, −2.010264514200981, −1.459546619716728, −0.8757002150115712, 0,
0.8757002150115712, 1.459546619716728, 2.010264514200981, 2.731886748347057, 3.361386741673103, 3.946244052695037, 4.648106469905472, 4.890164402488925, 5.722802635993704, 5.910991540161546, 6.407742440826966, 7.084967542341621, 7.563336531423349, 7.899808375852752, 8.657605521512483, 9.119627065197235, 9.481593602191633, 10.02902342488597, 10.57721095412866, 11.00283126662641, 11.33486807547579, 11.87541927437911, 12.22491645624514, 12.94808210357142, 13.26774098859784