| L(s) = 1 | − 2·2-s + 4·4-s − 12·5-s − 7·7-s − 8·8-s + 24·10-s − 60·11-s − 79·13-s + 14·14-s + 16·16-s + 108·17-s + 11·19-s − 48·20-s + 120·22-s + 132·23-s + 19·25-s + 158·26-s − 28·28-s − 96·29-s + 20·31-s − 32·32-s − 216·34-s + 84·35-s − 169·37-s − 22·38-s + 96·40-s − 192·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.07·5-s − 0.377·7-s − 0.353·8-s + 0.758·10-s − 1.64·11-s − 1.68·13-s + 0.267·14-s + 1/4·16-s + 1.54·17-s + 0.132·19-s − 0.536·20-s + 1.16·22-s + 1.19·23-s + 0.151·25-s + 1.19·26-s − 0.188·28-s − 0.614·29-s + 0.115·31-s − 0.176·32-s − 1.08·34-s + 0.405·35-s − 0.750·37-s − 0.0939·38-s + 0.379·40-s − 0.731·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 12 T + p^{3} T^{2} \) |
| 7 | \( 1 + p T + p^{3} T^{2} \) |
| 11 | \( 1 + 60 T + p^{3} T^{2} \) |
| 13 | \( 1 + 79 T + p^{3} T^{2} \) |
| 17 | \( 1 - 108 T + p^{3} T^{2} \) |
| 19 | \( 1 - 11 T + p^{3} T^{2} \) |
| 23 | \( 1 - 132 T + p^{3} T^{2} \) |
| 29 | \( 1 + 96 T + p^{3} T^{2} \) |
| 31 | \( 1 - 20 T + p^{3} T^{2} \) |
| 37 | \( 1 + 169 T + p^{3} T^{2} \) |
| 41 | \( 1 + 192 T + p^{3} T^{2} \) |
| 43 | \( 1 - 488 T + p^{3} T^{2} \) |
| 47 | \( 1 + 204 T + p^{3} T^{2} \) |
| 53 | \( 1 + 360 T + p^{3} T^{2} \) |
| 59 | \( 1 + 156 T + p^{3} T^{2} \) |
| 61 | \( 1 - 83 T + p^{3} T^{2} \) |
| 67 | \( 1 - 47 T + p^{3} T^{2} \) |
| 71 | \( 1 + 216 T + p^{3} T^{2} \) |
| 73 | \( 1 + 7 p T + p^{3} T^{2} \) |
| 79 | \( 1 + 529 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1128 T + p^{3} T^{2} \) |
| 89 | \( 1 + 36 T + p^{3} T^{2} \) |
| 97 | \( 1 - 605 T + p^{3} T^{2} \) |
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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
−14.64977708079262932133209378273, −12.83089366557875063372601960003, −11.95304218074722145216900810164, −10.63244531341256395370566892271, −9.604520174674355192397257396001, −7.939151097966394986534974661769, −7.33414166422753555699780592183, −5.18989312633925277632374299228, −2.99988311218391671495483567526, 0,
2.99988311218391671495483567526, 5.18989312633925277632374299228, 7.33414166422753555699780592183, 7.939151097966394986534974661769, 9.604520174674355192397257396001, 10.63244531341256395370566892271, 11.95304218074722145216900810164, 12.83089366557875063372601960003, 14.64977708079262932133209378273
