| L(s) = 1 | + 0.566·3-s − 3.92·5-s + 2.44·7-s − 2.67·9-s + 11-s + 6.76·13-s − 2.22·15-s − 0.175·17-s + 19-s + 1.38·21-s − 8.30·23-s + 10.4·25-s − 3.21·27-s − 0.843·29-s + 0.224·31-s + 0.566·33-s − 9.60·35-s − 3.49·37-s + 3.83·39-s + 5.09·41-s − 9.48·43-s + 10.5·45-s + 9.68·47-s − 1.01·49-s − 0.0996·51-s − 4.72·53-s − 3.92·55-s + ⋯ |
| L(s) = 1 | + 0.326·3-s − 1.75·5-s + 0.924·7-s − 0.893·9-s + 0.301·11-s + 1.87·13-s − 0.574·15-s − 0.0426·17-s + 0.229·19-s + 0.302·21-s − 1.73·23-s + 2.08·25-s − 0.618·27-s − 0.156·29-s + 0.0403·31-s + 0.0985·33-s − 1.62·35-s − 0.574·37-s + 0.613·39-s + 0.795·41-s − 1.44·43-s + 1.56·45-s + 1.41·47-s − 0.145·49-s − 0.0139·51-s − 0.649·53-s − 0.529·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.503652962\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.503652962\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 0.566T + 3T^{2} \) |
| 5 | \( 1 + 3.92T + 5T^{2} \) |
| 7 | \( 1 - 2.44T + 7T^{2} \) |
| 13 | \( 1 - 6.76T + 13T^{2} \) |
| 17 | \( 1 + 0.175T + 17T^{2} \) |
| 23 | \( 1 + 8.30T + 23T^{2} \) |
| 29 | \( 1 + 0.843T + 29T^{2} \) |
| 31 | \( 1 - 0.224T + 31T^{2} \) |
| 37 | \( 1 + 3.49T + 37T^{2} \) |
| 41 | \( 1 - 5.09T + 41T^{2} \) |
| 43 | \( 1 + 9.48T + 43T^{2} \) |
| 47 | \( 1 - 9.68T + 47T^{2} \) |
| 53 | \( 1 + 4.72T + 53T^{2} \) |
| 59 | \( 1 - 7.88T + 59T^{2} \) |
| 61 | \( 1 - 14.9T + 61T^{2} \) |
| 67 | \( 1 + 2.09T + 67T^{2} \) |
| 71 | \( 1 - 7.53T + 71T^{2} \) |
| 73 | \( 1 + 1.36T + 73T^{2} \) |
| 79 | \( 1 - 1.04T + 79T^{2} \) |
| 83 | \( 1 + 2.37T + 83T^{2} \) |
| 89 | \( 1 - 3.38T + 89T^{2} \) |
| 97 | \( 1 - 19.0T + 97T^{2} \) |
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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
−8.452745888600250497864495751296, −8.081243712570728569970925377743, −7.41476838932127469536544323534, −6.39975168066081403446645715351, −5.59987637402521134910031339263, −4.56891039344874644803739881293, −3.75424567039116553845107986065, −3.42814910990862365183226094700, −2.01482459983854826048500665619, −0.72428638465717122638185981304,
0.72428638465717122638185981304, 2.01482459983854826048500665619, 3.42814910990862365183226094700, 3.75424567039116553845107986065, 4.56891039344874644803739881293, 5.59987637402521134910031339263, 6.39975168066081403446645715351, 7.41476838932127469536544323534, 8.081243712570728569970925377743, 8.452745888600250497864495751296
