| L(s) = 1 | + 9.15·2-s + 105.·3-s − 428.·4-s − 1.75e3·5-s + 968.·6-s − 5.86e3·7-s − 8.60e3·8-s − 8.49e3·9-s − 1.60e4·10-s + 5.72e4·11-s − 4.52e4·12-s + 1.83e5·13-s − 5.37e4·14-s − 1.85e5·15-s + 1.40e5·16-s − 7.77e4·18-s − 2.69e5·19-s + 7.50e5·20-s − 6.20e5·21-s + 5.23e5·22-s + 1.82e6·23-s − 9.10e5·24-s + 1.11e6·25-s + 1.67e6·26-s − 2.98e6·27-s + 2.51e6·28-s − 1.33e6·29-s + ⋯ |
| L(s) = 1 | + 0.404·2-s + 0.753·3-s − 0.836·4-s − 1.25·5-s + 0.304·6-s − 0.924·7-s − 0.742·8-s − 0.431·9-s − 0.507·10-s + 1.17·11-s − 0.630·12-s + 1.77·13-s − 0.373·14-s − 0.945·15-s + 0.535·16-s − 0.174·18-s − 0.474·19-s + 1.04·20-s − 0.696·21-s + 0.476·22-s + 1.36·23-s − 0.560·24-s + 0.573·25-s + 0.720·26-s − 1.07·27-s + 0.772·28-s − 0.350·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 - 9.15T + 512T^{2} \) |
| 3 | \( 1 - 105.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.75e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 5.86e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 5.72e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.83e5T + 1.06e10T^{2} \) |
| 19 | \( 1 + 2.69e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.82e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.33e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 5.82e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.79e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.95e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.23e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.64e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 4.39e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 6.25e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.82e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.37e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 5.84e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.40e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.26e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 9.17e7T + 1.86e17T^{2} \) |
| 89 | \( 1 + 5.55e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 6.97e8T + 7.60e17T^{2} \) |
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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
−9.328956832333368073853638791785, −8.801493382866501892461518582467, −8.163165667046763352616680528225, −6.79403905817423888717584003755, −5.83055558114720698792907187380, −4.28752925792106498007430148120, −3.62592280863784882833709493271, −3.08791095009292911533607588541, −1.07942329505187014030870751742, 0,
1.07942329505187014030870751742, 3.08791095009292911533607588541, 3.62592280863784882833709493271, 4.28752925792106498007430148120, 5.83055558114720698792907187380, 6.79403905817423888717584003755, 8.163165667046763352616680528225, 8.801493382866501892461518582467, 9.328956832333368073853638791785
