| L(s) = 1 | + 2·2-s − 2·3-s + 4·4-s − 12·5-s − 4·6-s + 7·7-s + 8·8-s − 23·9-s − 24·10-s + 48·11-s − 8·12-s + 56·13-s + 14·14-s + 24·15-s + 16·16-s − 114·17-s − 46·18-s + 2·19-s − 48·20-s − 14·21-s + 96·22-s − 120·23-s − 16·24-s + 19·25-s + 112·26-s + 100·27-s + 28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.384·3-s + 1/2·4-s − 1.07·5-s − 0.272·6-s + 0.377·7-s + 0.353·8-s − 0.851·9-s − 0.758·10-s + 1.31·11-s − 0.192·12-s + 1.19·13-s + 0.267·14-s + 0.413·15-s + 1/4·16-s − 1.62·17-s − 0.602·18-s + 0.0241·19-s − 0.536·20-s − 0.145·21-s + 0.930·22-s − 1.08·23-s − 0.136·24-s + 0.151·25-s + 0.844·26-s + 0.712·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.136203385\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.136203385\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 7 | \( 1 - p T \) |
| good | 3 | \( 1 + 2 T + p^{3} T^{2} \) |
| 5 | \( 1 + 12 T + p^{3} T^{2} \) |
| 11 | \( 1 - 48 T + p^{3} T^{2} \) |
| 13 | \( 1 - 56 T + p^{3} T^{2} \) |
| 17 | \( 1 + 114 T + p^{3} T^{2} \) |
| 19 | \( 1 - 2 T + p^{3} T^{2} \) |
| 23 | \( 1 + 120 T + p^{3} T^{2} \) |
| 29 | \( 1 + 54 T + p^{3} T^{2} \) |
| 31 | \( 1 - 236 T + p^{3} T^{2} \) |
| 37 | \( 1 - 146 T + p^{3} T^{2} \) |
| 41 | \( 1 - 126 T + p^{3} T^{2} \) |
| 43 | \( 1 + 376 T + p^{3} T^{2} \) |
| 47 | \( 1 + 12 T + p^{3} T^{2} \) |
| 53 | \( 1 - 174 T + p^{3} T^{2} \) |
| 59 | \( 1 - 138 T + p^{3} T^{2} \) |
| 61 | \( 1 - 380 T + p^{3} T^{2} \) |
| 67 | \( 1 + 484 T + p^{3} T^{2} \) |
| 71 | \( 1 - 576 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1150 T + p^{3} T^{2} \) |
| 79 | \( 1 - 776 T + p^{3} T^{2} \) |
| 83 | \( 1 - 378 T + p^{3} T^{2} \) |
| 89 | \( 1 + 390 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1330 T + p^{3} T^{2} \) |
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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
−19.55131588676090196951499634617, −17.72055148312217158731195318477, −16.28749175378899646413987168836, −15.09686444946375758885149736496, −13.72439967177253770172734555098, −11.83139263343001526257239926146, −11.22780708054999787555304036773, −8.454298897419624859030526746703, −6.36455880219347397070988808395, −4.10728508949545943221697599589,
4.10728508949545943221697599589, 6.36455880219347397070988808395, 8.454298897419624859030526746703, 11.22780708054999787555304036773, 11.83139263343001526257239926146, 13.72439967177253770172734555098, 15.09686444946375758885149736496, 16.28749175378899646413987168836, 17.72055148312217158731195318477, 19.55131588676090196951499634617
