| L(s) = 1 | − 3.65·5-s − 2.28·7-s − 3.94·11-s + 4.52·13-s − 2·17-s + 2.83·19-s + 5.83·23-s + 8.35·25-s + 0.0496·29-s + 2.92·31-s + 8.36·35-s + 37-s + 10.1·41-s + 3.31·43-s − 13.1·47-s − 1.76·49-s + 0.188·53-s + 14.4·55-s + 10.7·59-s + 6.96·61-s − 16.5·65-s + 1.17·67-s − 14.5·71-s − 12.5·73-s + 9.02·77-s − 14.7·79-s − 2.38·83-s + ⋯ |
| L(s) = 1 | − 1.63·5-s − 0.864·7-s − 1.18·11-s + 1.25·13-s − 0.485·17-s + 0.650·19-s + 1.21·23-s + 1.67·25-s + 0.00922·29-s + 0.524·31-s + 1.41·35-s + 0.164·37-s + 1.58·41-s + 0.504·43-s − 1.91·47-s − 0.252·49-s + 0.0259·53-s + 1.94·55-s + 1.39·59-s + 0.891·61-s − 2.05·65-s + 0.143·67-s − 1.72·71-s − 1.47·73-s + 1.02·77-s − 1.66·79-s − 0.262·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 5 | \( 1 + 3.65T + 5T^{2} \) |
| 7 | \( 1 + 2.28T + 7T^{2} \) |
| 11 | \( 1 + 3.94T + 11T^{2} \) |
| 13 | \( 1 - 4.52T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 2.83T + 19T^{2} \) |
| 23 | \( 1 - 5.83T + 23T^{2} \) |
| 29 | \( 1 - 0.0496T + 29T^{2} \) |
| 31 | \( 1 - 2.92T + 31T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 - 3.31T + 43T^{2} \) |
| 47 | \( 1 + 13.1T + 47T^{2} \) |
| 53 | \( 1 - 0.188T + 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 - 6.96T + 61T^{2} \) |
| 67 | \( 1 - 1.17T + 67T^{2} \) |
| 71 | \( 1 + 14.5T + 71T^{2} \) |
| 73 | \( 1 + 12.5T + 73T^{2} \) |
| 79 | \( 1 + 14.7T + 79T^{2} \) |
| 83 | \( 1 + 2.38T + 83T^{2} \) |
| 89 | \( 1 - 13.4T + 89T^{2} \) |
| 97 | \( 1 + 1.62T + 97T^{2} \) |
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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
−7.80624344339939031271754949761, −7.22835653559981074457560133695, −6.52381187309632108675556590592, −5.66890802359654147246327479690, −4.76448611699728315392996468639, −4.03616776795219070226034523524, −3.25819910207246213941496816681, −2.75038538302988851460966929268, −1.03671774653865291159955845796, 0,
1.03671774653865291159955845796, 2.75038538302988851460966929268, 3.25819910207246213941496816681, 4.03616776795219070226034523524, 4.76448611699728315392996468639, 5.66890802359654147246327479690, 6.52381187309632108675556590592, 7.22835653559981074457560133695, 7.80624344339939031271754949761
