| L(s) = 1 | − 29·3-s + 25·5-s + 259·7-s + 243·9-s + 718·11-s + 1.97e3·13-s − 725·15-s + 528·17-s − 2.39e3·19-s − 7.51e3·21-s + 743·23-s + 3.24e3·27-s + 1.65e3·29-s + 3.27e3·31-s − 2.08e4·33-s + 6.47e3·35-s − 4.31e3·37-s − 5.73e4·39-s + 3.45e4·41-s − 2.85e4·43-s + 6.07e3·45-s − 1.05e4·47-s + 5.02e4·49-s − 1.53e4·51-s − 3.80e3·53-s + 1.79e4·55-s + 6.94e4·57-s + ⋯ |
| L(s) = 1 | − 1.86·3-s + 0.447·5-s + 1.99·7-s + 9-s + 1.78·11-s + 3.24·13-s − 0.831·15-s + 0.443·17-s − 1.52·19-s − 3.71·21-s + 0.292·23-s + 0.857·27-s + 0.366·29-s + 0.611·31-s − 3.32·33-s + 0.893·35-s − 0.518·37-s − 6.03·39-s + 3.21·41-s − 2.35·43-s + 0.447·45-s − 0.693·47-s + 2.99·49-s − 0.824·51-s − 0.185·53-s + 0.800·55-s + 2.83·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.800991455\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.800991455\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - p^{2} T + p^{4} T^{2} \) |
| 7 | $C_2$ | \( 1 - 37 p T + p^{5} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 29 T + 598 T^{2} + 29 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 718 T + 354473 T^{2} - 718 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 76 p T + p^{5} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 528 T - 1141073 T^{2} - 528 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 126 p T + 9017 p^{2} T^{2} + 126 p^{6} T^{3} + p^{10} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 743 T - 5884294 T^{2} - 743 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 829 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 3272 T - 17923167 T^{2} - 3272 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 4316 T - 50716101 T^{2} + 4316 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 17281 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 14251 T + p^{5} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 10504 T - 119010991 T^{2} + 10504 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 3802 T - 403740289 T^{2} + 3802 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 15874 T - 462940423 T^{2} + 15874 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 27069 T - 111865540 T^{2} + 27069 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 28503 T - 537704098 T^{2} - 28503 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 49230 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 20496 T - 1652985577 T^{2} + 20496 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 37238 T - 1690387755 T^{2} - 37238 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 22177 T + p^{5} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 43025 T - 3732908824 T^{2} + 43025 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 152030 T + p^{5} T^{2} )^{2} \) |
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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
−12.24630501681202980275411695031, −11.62750348162410976002697035461, −11.45334598116718221592452185605, −11.15904309291239522389744355413, −10.65690976986397815836973396613, −10.42624085493214960426391203265, −9.236268937151348019549985979241, −8.633267843593482558751427678448, −8.537733262252079763577426458582, −7.79800696755396898831724579413, −6.62408914194630189396831754778, −6.41736192755680576053948394558, −5.81457509843285648764446134320, −5.63197004985791723615666867419, −4.45914742985200815356101882213, −4.39539936805208043110235504953, −3.37756796083528342467040540909, −1.75423548553048908485691011825, −1.29639900681023814903835895968, −0.78554986349373304927531678479,
0.78554986349373304927531678479, 1.29639900681023814903835895968, 1.75423548553048908485691011825, 3.37756796083528342467040540909, 4.39539936805208043110235504953, 4.45914742985200815356101882213, 5.63197004985791723615666867419, 5.81457509843285648764446134320, 6.41736192755680576053948394558, 6.62408914194630189396831754778, 7.79800696755396898831724579413, 8.537733262252079763577426458582, 8.633267843593482558751427678448, 9.236268937151348019549985979241, 10.42624085493214960426391203265, 10.65690976986397815836973396613, 11.15904309291239522389744355413, 11.45334598116718221592452185605, 11.62750348162410976002697035461, 12.24630501681202980275411695031
