| L(s) = 1 | − 5-s + 4·7-s − 3·9-s + 4·13-s − 4·17-s + 8·19-s + 4·23-s + 25-s − 8·29-s − 4·31-s − 4·35-s + 6·37-s − 8·41-s + 4·43-s + 3·45-s + 12·47-s + 9·49-s − 10·53-s + 8·61-s − 12·63-s − 4·65-s + 8·67-s + 12·71-s + 12·73-s − 8·79-s + 9·81-s + 4·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.51·7-s − 9-s + 1.10·13-s − 0.970·17-s + 1.83·19-s + 0.834·23-s + 1/5·25-s − 1.48·29-s − 0.718·31-s − 0.676·35-s + 0.986·37-s − 1.24·41-s + 0.609·43-s + 0.447·45-s + 1.75·47-s + 9/7·49-s − 1.37·53-s + 1.02·61-s − 1.51·63-s − 0.496·65-s + 0.977·67-s + 1.42·71-s + 1.40·73-s − 0.900·79-s + 81-s + 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.947074675\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.947074675\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.897134461504400081047923465652, −8.128357942356163602517691543555, −7.63730249514195677328277080991, −6.72371044468655346592672834809, −5.55512436880508977002270669134, −5.18314326934068116473020730219, −4.09939457313647273796966014465, −3.27168532579693518607528405834, −2.08585037934978704873298400028, −0.932352676039322010550134153758,
0.932352676039322010550134153758, 2.08585037934978704873298400028, 3.27168532579693518607528405834, 4.09939457313647273796966014465, 5.18314326934068116473020730219, 5.55512436880508977002270669134, 6.72371044468655346592672834809, 7.63730249514195677328277080991, 8.128357942356163602517691543555, 8.897134461504400081047923465652
