| L(s) = 1 | + 1.94·3-s − 0.422·5-s − 0.503·7-s + 0.777·9-s + 2.53·11-s − 0.681·13-s − 0.820·15-s − 5.42·17-s − 0.978·21-s + 3.01·23-s − 4.82·25-s − 4.31·27-s + 7.18·29-s − 10.4·31-s + 4.91·33-s + 0.212·35-s − 4.38·37-s − 1.32·39-s − 9.34·41-s + 4.16·43-s − 0.328·45-s − 10.2·47-s − 6.74·49-s − 10.5·51-s + 4.18·53-s − 1.06·55-s + 9.17·59-s + ⋯ |
| L(s) = 1 | + 1.12·3-s − 0.188·5-s − 0.190·7-s + 0.259·9-s + 0.762·11-s − 0.189·13-s − 0.211·15-s − 1.31·17-s − 0.213·21-s + 0.629·23-s − 0.964·25-s − 0.831·27-s + 1.33·29-s − 1.87·31-s + 0.856·33-s + 0.0359·35-s − 0.721·37-s − 0.212·39-s − 1.45·41-s + 0.635·43-s − 0.0489·45-s − 1.48·47-s − 0.963·49-s − 1.47·51-s + 0.574·53-s − 0.144·55-s + 1.19·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 1.94T + 3T^{2} \) |
| 5 | \( 1 + 0.422T + 5T^{2} \) |
| 7 | \( 1 + 0.503T + 7T^{2} \) |
| 11 | \( 1 - 2.53T + 11T^{2} \) |
| 13 | \( 1 + 0.681T + 13T^{2} \) |
| 17 | \( 1 + 5.42T + 17T^{2} \) |
| 23 | \( 1 - 3.01T + 23T^{2} \) |
| 29 | \( 1 - 7.18T + 29T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 37 | \( 1 + 4.38T + 37T^{2} \) |
| 41 | \( 1 + 9.34T + 41T^{2} \) |
| 43 | \( 1 - 4.16T + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 - 4.18T + 53T^{2} \) |
| 59 | \( 1 - 9.17T + 59T^{2} \) |
| 61 | \( 1 - 0.515T + 61T^{2} \) |
| 67 | \( 1 + 7.91T + 67T^{2} \) |
| 71 | \( 1 - 2.98T + 71T^{2} \) |
| 73 | \( 1 - 10.7T + 73T^{2} \) |
| 79 | \( 1 - 2.21T + 79T^{2} \) |
| 83 | \( 1 + 1.59T + 83T^{2} \) |
| 89 | \( 1 + 12.8T + 89T^{2} \) |
| 97 | \( 1 + 3.47T + 97T^{2} \) |
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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
−7.934166415456537022448381921332, −6.99328574932733206831296701324, −6.63221132053116867913513906701, −5.58138829119809569968790547009, −4.70203880360824070967361603233, −3.83811316569775782828342785794, −3.31059181753084243031065632452, −2.37110136490120538811096112190, −1.61958593771060691584990875736, 0,
1.61958593771060691584990875736, 2.37110136490120538811096112190, 3.31059181753084243031065632452, 3.83811316569775782828342785794, 4.70203880360824070967361603233, 5.58138829119809569968790547009, 6.63221132053116867913513906701, 6.99328574932733206831296701324, 7.934166415456537022448381921332
