| L(s) = 1 | + 3·3-s + 2·5-s − 2·7-s + 6·9-s + 32·11-s + 27·13-s + 6·15-s − 22·17-s + 38·19-s − 6·21-s − 40·23-s + 25·25-s + 9·27-s + 30·29-s + 96·33-s − 4·35-s + 81·39-s − 24·41-s + 49·43-s + 12·45-s − 46·47-s − 95·49-s − 66·51-s − 84·53-s + 64·55-s + 114·57-s − 114·59-s + ⋯ |
| L(s) = 1 | + 3-s + 2/5·5-s − 2/7·7-s + 2/3·9-s + 2.90·11-s + 2.07·13-s + 2/5·15-s − 1.29·17-s + 2·19-s − 2/7·21-s − 1.73·23-s + 25-s + 1/3·27-s + 1.03·29-s + 2.90·33-s − 0.114·35-s + 2.07·39-s − 0.585·41-s + 1.13·43-s + 4/15·45-s − 0.978·47-s − 1.93·49-s − 1.29·51-s − 1.58·53-s + 1.16·55-s + 2·57-s − 1.93·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51984 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(4.116687190\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.116687190\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 19 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T - 21 T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 16 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 27 T + 412 T^{2} - 27 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 22 T + 195 T^{2} + 22 p^{2} T^{3} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 40 T + 1071 T^{2} + 40 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 30 T + 1141 T^{2} - 30 p^{2} T^{3} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 601 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2495 T^{2} + p^{4} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 24 T + 1873 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 49 T + 552 T^{2} - 49 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 46 T - 93 T^{2} + 46 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 84 T + 5161 T^{2} + 84 p^{2} T^{3} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 114 T + 7813 T^{2} + 114 p^{2} T^{3} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 97 T + 5688 T^{2} - 97 p^{2} T^{3} + p^{4} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 45 T + 5164 T^{2} + 45 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 84 T + 7393 T^{2} + 84 p^{2} T^{3} + p^{4} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 35 T - 4104 T^{2} + 35 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 11 T + p^{2} T^{2} )( 1 + 142 T + p^{2} T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 146 T + p^{2} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 66 T + 9373 T^{2} + 66 p^{2} T^{3} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 108 T + 13297 T^{2} - 108 p^{2} T^{3} + p^{4} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.10931683149161927560531041171, −11.73340007688584468668695380825, −11.34797008395468596471422866420, −10.92457264940197545839318516757, −10.04043919725702964297232537386, −9.680938032547247718881869956298, −9.243310308260031291279739273810, −8.868622697071046235476206118518, −8.446064498246633033403258436856, −8.002925463593753517517403096494, −6.88657719691798601487928393466, −6.85021764243956778025087138377, −6.09117126068293478539030208026, −5.87244992220180408910093251030, −4.38081014211925917088196726164, −4.31605659965606480652765252899, −3.33680341240190845421594917441, −3.12420560569280445443122636641, −1.47063226562894192984718442758, −1.46304830301414513428026097913,
1.46304830301414513428026097913, 1.47063226562894192984718442758, 3.12420560569280445443122636641, 3.33680341240190845421594917441, 4.31605659965606480652765252899, 4.38081014211925917088196726164, 5.87244992220180408910093251030, 6.09117126068293478539030208026, 6.85021764243956778025087138377, 6.88657719691798601487928393466, 8.002925463593753517517403096494, 8.446064498246633033403258436856, 8.868622697071046235476206118518, 9.243310308260031291279739273810, 9.680938032547247718881869956298, 10.04043919725702964297232537386, 10.92457264940197545839318516757, 11.34797008395468596471422866420, 11.73340007688584468668695380825, 12.10931683149161927560531041171
