| L(s) = 1 | + 2·3-s + 34·5-s + 20·7-s + 27·9-s − 32·11-s − 91·13-s + 68·15-s + 13·17-s + 30·19-s + 40·21-s + 78·23-s + 617·25-s + 154·27-s − 197·29-s + 148·31-s − 64·33-s + 680·35-s + 227·37-s − 182·39-s + 165·41-s − 156·43-s + 918·45-s + 324·47-s + 343·49-s + 26·51-s + 186·53-s − 1.08e3·55-s + ⋯ |
| L(s) = 1 | + 0.384·3-s + 3.04·5-s + 1.07·7-s + 9-s − 0.877·11-s − 1.94·13-s + 1.17·15-s + 0.185·17-s + 0.362·19-s + 0.415·21-s + 0.707·23-s + 4.93·25-s + 1.09·27-s − 1.26·29-s + 0.857·31-s − 0.337·33-s + 3.28·35-s + 1.00·37-s − 0.747·39-s + 0.628·41-s − 0.553·43-s + 3.04·45-s + 1.00·47-s + 49-s + 0.0713·51-s + 0.482·53-s − 2.66·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43264 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43264 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.082947577\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.082947577\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + 7 p T + p^{3} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T - 23 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 17 T + p^{3} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 37 T + p^{3} T^{2} )( 1 + 17 T + p^{3} T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 32 T - 307 T^{2} + 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 13 T - 4744 T^{2} - 13 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 30 T - 5959 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 78 T - 6083 T^{2} - 78 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 197 T + 14420 T^{2} + 197 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 74 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 227 T + 876 T^{2} - 227 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 165 T - 41696 T^{2} - 165 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 156 T - 55171 T^{2} + 156 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 162 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 93 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 864 T + 541117 T^{2} + 864 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 145 T - 205956 T^{2} + 145 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 862 T + 442281 T^{2} - 862 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 654 T + 69805 T^{2} - 654 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 215 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 76 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 628 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 266 T - 634213 T^{2} - 266 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 238 T - 856029 T^{2} + 238 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−12.42888366832277927231079061537, −11.77516542379583694993870160150, −10.85199155655334125232058488297, −10.63530718574533131156556485095, −10.00213732864472195410405770145, −9.824770643308559952565446914193, −9.230375047623938384304385872606, −9.171694638666166837257327746599, −7.991301436654940920483079581914, −7.78477331475611369065797644783, −6.92656963582922957328894122316, −6.64793482900102370352900460143, −5.55469201608249435461346792030, −5.51045502909727910653454318104, −4.91948698875019801217357943405, −4.34618637320747335443641024291, −2.74162376534802572463306772283, −2.47754328771313182123759651992, −1.83185941458529089643645018017, −1.13716571263080015037570963466,
1.13716571263080015037570963466, 1.83185941458529089643645018017, 2.47754328771313182123759651992, 2.74162376534802572463306772283, 4.34618637320747335443641024291, 4.91948698875019801217357943405, 5.51045502909727910653454318104, 5.55469201608249435461346792030, 6.64793482900102370352900460143, 6.92656963582922957328894122316, 7.78477331475611369065797644783, 7.991301436654940920483079581914, 9.171694638666166837257327746599, 9.230375047623938384304385872606, 9.824770643308559952565446914193, 10.00213732864472195410405770145, 10.63530718574533131156556485095, 10.85199155655334125232058488297, 11.77516542379583694993870160150, 12.42888366832277927231079061537
