| L(s) = 1 | + 3.37·3-s + 5-s + 3.37·7-s + 8.37·9-s − 2·13-s + 3.37·15-s − 1.37·17-s + 0.627·19-s + 11.3·21-s − 2.74·23-s + 25-s + 18.1·27-s − 1.37·29-s − 3.37·31-s + 3.37·35-s + 9.37·37-s − 6.74·39-s + 11.4·41-s − 4·43-s + 8.37·45-s − 2.74·47-s + 4.37·49-s − 4.62·51-s − 4.11·53-s + 2.11·57-s + 2.74·59-s + 5.37·61-s + ⋯ |
| L(s) = 1 | + 1.94·3-s + 0.447·5-s + 1.27·7-s + 2.79·9-s − 0.554·13-s + 0.870·15-s − 0.332·17-s + 0.144·19-s + 2.48·21-s − 0.572·23-s + 0.200·25-s + 3.48·27-s − 0.254·29-s − 0.605·31-s + 0.570·35-s + 1.54·37-s − 1.07·39-s + 1.79·41-s − 0.609·43-s + 1.24·45-s − 0.400·47-s + 0.624·49-s − 0.648·51-s − 0.565·53-s + 0.280·57-s + 0.357·59-s + 0.687·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.229379805\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.229379805\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.37T + 3T^{2} \) |
| 7 | \( 1 - 3.37T + 7T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 1.37T + 17T^{2} \) |
| 19 | \( 1 - 0.627T + 19T^{2} \) |
| 23 | \( 1 + 2.74T + 23T^{2} \) |
| 29 | \( 1 + 1.37T + 29T^{2} \) |
| 31 | \( 1 + 3.37T + 31T^{2} \) |
| 37 | \( 1 - 9.37T + 37T^{2} \) |
| 41 | \( 1 - 11.4T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 2.74T + 47T^{2} \) |
| 53 | \( 1 + 4.11T + 53T^{2} \) |
| 59 | \( 1 - 2.74T + 59T^{2} \) |
| 61 | \( 1 - 5.37T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 - 15.4T + 73T^{2} \) |
| 79 | \( 1 + 1.25T + 79T^{2} \) |
| 83 | \( 1 + 2.74T + 83T^{2} \) |
| 89 | \( 1 + 1.37T + 89T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
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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
−7.70837678115078068468307058856, −7.42443082449119976594260592075, −6.51873327884104117883541925431, −5.52347934693889609092478165812, −4.58740077518248339102837076369, −4.23678192317622752818382017896, −3.28622364632750407140107091290, −2.44858886070024189818870757787, −1.99662104973823714369183526767, −1.18762964365922454785456621733,
1.18762964365922454785456621733, 1.99662104973823714369183526767, 2.44858886070024189818870757787, 3.28622364632750407140107091290, 4.23678192317622752818382017896, 4.58740077518248339102837076369, 5.52347934693889609092478165812, 6.51873327884104117883541925431, 7.42443082449119976594260592075, 7.70837678115078068468307058856
