| L(s) = 1 | + 2-s + 2·3-s + 2·6-s − 7-s − 8-s + 3·9-s + 13-s − 14-s − 16-s + 3·18-s − 2·21-s − 2·23-s − 2·24-s + 25-s + 26-s + 4·27-s + 31-s − 37-s + 2·39-s − 41-s − 2·42-s − 2·43-s − 2·46-s − 2·48-s + 50-s − 53-s + 4·54-s + ⋯ |
| L(s) = 1 | + 2-s + 2·3-s + 2·6-s − 7-s − 8-s + 3·9-s + 13-s − 14-s − 16-s + 3·18-s − 2·21-s − 2·23-s − 2·24-s + 25-s + 26-s + 4·27-s + 31-s − 37-s + 2·39-s − 41-s − 2·42-s − 2·43-s − 2·46-s − 2·48-s + 50-s − 53-s + 4·54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.068330778\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.068330778\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 2 | \( 1 - T + T^{2} \) |
| 3 | \( ( 1 - T )^{2} \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( 1 - T + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 + T )^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 - T + T^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( ( 1 + T )^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 + T + T^{2} \) |
| 61 | \( 1 - T + T^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( 1 - T + T^{2} \) |
| 97 | \( 1 - T + T^{2} \) |
| 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
−9.074349222507728275817886622518, −8.531735011397054327379470639631, −7.947287248199407782335837952722, −6.72881383777803676283419226759, −6.31212754378585160821104145666, −4.93393645642217672891214012793, −4.05688685730382910616889281673, −3.42204136060942317669847653260, −2.96889043660624320916559542723, −1.78534481204312930370271025218,
1.78534481204312930370271025218, 2.96889043660624320916559542723, 3.42204136060942317669847653260, 4.05688685730382910616889281673, 4.93393645642217672891214012793, 6.31212754378585160821104145666, 6.72881383777803676283419226759, 7.947287248199407782335837952722, 8.531735011397054327379470639631, 9.074349222507728275817886622518
