| L(s) = 1 | + 2.68·3-s − 3.29·5-s − 2.35·7-s + 4.20·9-s + 1.21·13-s − 8.85·15-s − 0.522·17-s − 19-s − 6.31·21-s + 6.95·23-s + 5.88·25-s + 3.22·27-s + 0.192·29-s + 7.00·31-s + 7.76·35-s − 6.79·37-s + 3.25·39-s − 5.91·41-s − 1.42·43-s − 13.8·45-s + 0.946·47-s − 1.46·49-s − 1.40·51-s − 1.83·53-s − 2.68·57-s − 0.306·59-s − 1.24·61-s + ⋯ |
| L(s) = 1 | + 1.54·3-s − 1.47·5-s − 0.889·7-s + 1.40·9-s + 0.335·13-s − 2.28·15-s − 0.126·17-s − 0.229·19-s − 1.37·21-s + 1.44·23-s + 1.17·25-s + 0.620·27-s + 0.0357·29-s + 1.25·31-s + 1.31·35-s − 1.11·37-s + 0.520·39-s − 0.924·41-s − 0.218·43-s − 2.06·45-s + 0.138·47-s − 0.209·49-s − 0.196·51-s − 0.252·53-s − 0.355·57-s − 0.0398·59-s − 0.159·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 2.68T + 3T^{2} \) |
| 5 | \( 1 + 3.29T + 5T^{2} \) |
| 7 | \( 1 + 2.35T + 7T^{2} \) |
| 13 | \( 1 - 1.21T + 13T^{2} \) |
| 17 | \( 1 + 0.522T + 17T^{2} \) |
| 23 | \( 1 - 6.95T + 23T^{2} \) |
| 29 | \( 1 - 0.192T + 29T^{2} \) |
| 31 | \( 1 - 7.00T + 31T^{2} \) |
| 37 | \( 1 + 6.79T + 37T^{2} \) |
| 41 | \( 1 + 5.91T + 41T^{2} \) |
| 43 | \( 1 + 1.42T + 43T^{2} \) |
| 47 | \( 1 - 0.946T + 47T^{2} \) |
| 53 | \( 1 + 1.83T + 53T^{2} \) |
| 59 | \( 1 + 0.306T + 59T^{2} \) |
| 61 | \( 1 + 1.24T + 61T^{2} \) |
| 67 | \( 1 - 1.36T + 67T^{2} \) |
| 71 | \( 1 - 2.99T + 71T^{2} \) |
| 73 | \( 1 + 16.4T + 73T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 - 4.74T + 83T^{2} \) |
| 89 | \( 1 + 11.6T + 89T^{2} \) |
| 97 | \( 1 + 16.9T + 97T^{2} \) |
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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
−7.40429026337958739133943246987, −7.00380602175869989604356695260, −6.29716505435352995380694123157, −5.04102335954591202383957901414, −4.30574663255239631577814715137, −3.56333714999103952349493704732, −3.21767215509619826869562647340, −2.53578324232743137566296023863, −1.29055174636256270175702237489, 0,
1.29055174636256270175702237489, 2.53578324232743137566296023863, 3.21767215509619826869562647340, 3.56333714999103952349493704732, 4.30574663255239631577814715137, 5.04102335954591202383957901414, 6.29716505435352995380694123157, 7.00380602175869989604356695260, 7.40429026337958739133943246987
