| L(s) = 1 | − 0.426·2-s − 3-s − 1.81·4-s − 3.71·5-s + 0.426·6-s + 0.191·7-s + 1.62·8-s + 9-s + 1.58·10-s + 1.81·12-s − 3.42·13-s − 0.0816·14-s + 3.71·15-s + 2.94·16-s − 3.01·17-s − 0.426·18-s − 19-s + 6.76·20-s − 0.191·21-s − 3.58·23-s − 1.62·24-s + 8.82·25-s + 1.45·26-s − 27-s − 0.348·28-s − 3.86·29-s − 1.58·30-s + ⋯ |
| L(s) = 1 | − 0.301·2-s − 0.577·3-s − 0.909·4-s − 1.66·5-s + 0.173·6-s + 0.0724·7-s + 0.575·8-s + 0.333·9-s + 0.500·10-s + 0.524·12-s − 0.950·13-s − 0.0218·14-s + 0.960·15-s + 0.735·16-s − 0.730·17-s − 0.100·18-s − 0.229·19-s + 1.51·20-s − 0.0418·21-s − 0.747·23-s − 0.332·24-s + 1.76·25-s + 0.286·26-s − 0.192·27-s − 0.0658·28-s − 0.716·29-s − 0.289·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6897 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6897 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.02177358767\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02177358767\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 0.426T + 2T^{2} \) |
| 5 | \( 1 + 3.71T + 5T^{2} \) |
| 7 | \( 1 - 0.191T + 7T^{2} \) |
| 13 | \( 1 + 3.42T + 13T^{2} \) |
| 17 | \( 1 + 3.01T + 17T^{2} \) |
| 23 | \( 1 + 3.58T + 23T^{2} \) |
| 29 | \( 1 + 3.86T + 29T^{2} \) |
| 31 | \( 1 - 3.39T + 31T^{2} \) |
| 37 | \( 1 + 5.08T + 37T^{2} \) |
| 41 | \( 1 - 5.96T + 41T^{2} \) |
| 43 | \( 1 + 9.30T + 43T^{2} \) |
| 47 | \( 1 + 4.37T + 47T^{2} \) |
| 53 | \( 1 - 12.5T + 53T^{2} \) |
| 59 | \( 1 - 9.02T + 59T^{2} \) |
| 61 | \( 1 + 9.90T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 + 8.13T + 71T^{2} \) |
| 73 | \( 1 + 13.1T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 + 0.914T + 83T^{2} \) |
| 89 | \( 1 + 13.9T + 89T^{2} \) |
| 97 | \( 1 + 6.26T + 97T^{2} \) |
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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
−7.989595099640236215298188775839, −7.35865954214023416207600656032, −6.81699201298416767858096268158, −5.72306434650964665755228863800, −4.92236103200635012152442096291, −4.33433446009497026850606031691, −3.91344152567999103615128947575, −2.88127887758078494003915472312, −1.47689912986574635175606327582, −0.084839586329581087123770546192,
0.084839586329581087123770546192, 1.47689912986574635175606327582, 2.88127887758078494003915472312, 3.91344152567999103615128947575, 4.33433446009497026850606031691, 4.92236103200635012152442096291, 5.72306434650964665755228863800, 6.81699201298416767858096268158, 7.35865954214023416207600656032, 7.989595099640236215298188775839
