| L(s) = 1 | + 2·5-s − 5·7-s − 5·11-s − 2·13-s − 17-s + 3·19-s + 3·23-s + 3·25-s − 29-s − 10·35-s − 7·37-s − 5·41-s − 5·43-s − 24·47-s + 7·49-s − 4·53-s − 10·55-s − 11·59-s + 13·61-s − 4·65-s − 3·67-s + 13·71-s − 4·73-s + 25·77-s − 8·79-s − 24·83-s − 2·85-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.88·7-s − 1.50·11-s − 0.554·13-s − 0.242·17-s + 0.688·19-s + 0.625·23-s + 3/5·25-s − 0.185·29-s − 1.69·35-s − 1.15·37-s − 0.780·41-s − 0.762·43-s − 3.50·47-s + 49-s − 0.549·53-s − 1.34·55-s − 1.43·59-s + 1.66·61-s − 0.496·65-s − 0.366·67-s + 1.54·71-s − 0.468·73-s + 2.84·77-s − 0.900·79-s − 2.63·83-s − 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5475600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5475600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + T - 16 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 3 T - 10 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + T - 28 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 5 T - 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 11 T + 62 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 3 T - 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 13 T + 98 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 7 T - 40 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 11 T + 24 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
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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
−8.838814563167427910912216395617, −8.434545929684705605988652417415, −8.101544759683178480942056653529, −7.60496446526322842336959090367, −7.03944542234298303346900903577, −6.92655763290294997094913865823, −6.40490747236835220965002001196, −6.22584556575060617728076313024, −5.59110309363493678302391583515, −5.24794756168377017092136742792, −4.95030696853527822737021295000, −4.58659774407453874784089739177, −3.54267025557869565167691520184, −3.45818535933561130442034590402, −2.80624486224417120294357100751, −2.74000725852010898721348922002, −1.89729501922023660337250120545, −1.37797793068680699313953935551, 0, 0,
1.37797793068680699313953935551, 1.89729501922023660337250120545, 2.74000725852010898721348922002, 2.80624486224417120294357100751, 3.45818535933561130442034590402, 3.54267025557869565167691520184, 4.58659774407453874784089739177, 4.95030696853527822737021295000, 5.24794756168377017092136742792, 5.59110309363493678302391583515, 6.22584556575060617728076313024, 6.40490747236835220965002001196, 6.92655763290294997094913865823, 7.03944542234298303346900903577, 7.60496446526322842336959090367, 8.101544759683178480942056653529, 8.434545929684705605988652417415, 8.838814563167427910912216395617
