| L(s) = 1 | − 5·5-s + 24.8·7-s + 25.7·11-s + 60.6·13-s + 28.6·17-s + 86.6·19-s + 52.3·23-s + 25·25-s − 6·29-s + 84.8·31-s − 124.·35-s − 448.·37-s − 183.·41-s + 252·43-s − 41.9·47-s + 275.·49-s + 228.·53-s − 128.·55-s − 179.·59-s + 480.·61-s − 303.·65-s + 855.·67-s − 675.·71-s − 621.·73-s + 640·77-s − 513.·79-s − 1.28e3·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.34·7-s + 0.705·11-s + 1.29·13-s + 0.408·17-s + 1.04·19-s + 0.474·23-s + 0.200·25-s − 0.0384·29-s + 0.491·31-s − 0.600·35-s − 1.99·37-s − 0.697·41-s + 0.893·43-s − 0.130·47-s + 0.802·49-s + 0.592·53-s − 0.315·55-s − 0.396·59-s + 1.00·61-s − 0.578·65-s + 1.55·67-s − 1.12·71-s − 0.996·73-s + 0.947·77-s − 0.731·79-s − 1.69·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.950876777\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.950876777\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| good | 7 | \( 1 - 24.8T + 343T^{2} \) |
| 11 | \( 1 - 25.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 60.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 28.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 86.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 52.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 84.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + 448.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 183.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 252T + 7.95e4T^{2} \) |
| 47 | \( 1 + 41.9T + 1.03e5T^{2} \) |
| 53 | \( 1 - 228.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 179.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 480.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 855.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 675.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 621.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 513.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.28e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.00e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 300.T + 9.12e5T^{2} \) |
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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
−8.851343842952510226479894171857, −8.486259511302156169554758653383, −7.59087707230108840799453744459, −6.86238698238456926427953602609, −5.75105929495843700690488382764, −4.97928627411814365627594945927, −4.02770528711140308338417777794, −3.21953989869756995939816356797, −1.68834627582591860309520997237, −0.942637270778345035700060829973,
0.942637270778345035700060829973, 1.68834627582591860309520997237, 3.21953989869756995939816356797, 4.02770528711140308338417777794, 4.97928627411814365627594945927, 5.75105929495843700690488382764, 6.86238698238456926427953602609, 7.59087707230108840799453744459, 8.486259511302156169554758653383, 8.851343842952510226479894171857
