| L(s) = 1 | − 8·5-s − 30·7-s + 46·11-s + 92·13-s − 22·17-s − 86·19-s + 58·23-s − 25-s − 108·29-s + 136·31-s + 240·35-s − 180·37-s − 672·41-s − 70·43-s − 852·47-s + 190·49-s − 668·53-s − 368·55-s − 548·59-s + 284·61-s − 736·65-s + 1.16e3·67-s + 806·71-s − 1.51e3·73-s − 1.38e3·77-s − 22·79-s − 1.54e3·83-s + ⋯ |
| L(s) = 1 | − 0.715·5-s − 1.61·7-s + 1.26·11-s + 1.96·13-s − 0.313·17-s − 1.03·19-s + 0.525·23-s − 0.00799·25-s − 0.691·29-s + 0.787·31-s + 1.15·35-s − 0.799·37-s − 2.55·41-s − 0.248·43-s − 2.64·47-s + 0.553·49-s − 1.73·53-s − 0.902·55-s − 1.20·59-s + 0.596·61-s − 1.40·65-s + 2.11·67-s + 1.34·71-s − 2.42·73-s − 2.04·77-s − 0.0313·79-s − 2.03·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 419904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 419904 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 + 8 T + 13 p T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 30 T + 710 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 46 T + 1382 T^{2} - 46 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 92 T + 6309 T^{2} - 92 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 22 T - 2917 T^{2} + 22 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 86 T + 10542 T^{2} + 86 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 58 T + 24974 T^{2} - 58 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 108 T + 51493 T^{2} + 108 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 136 T + 12750 T^{2} - 136 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 180 T + 109205 T^{2} + 180 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 672 T + 249934 T^{2} + 672 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 70 T + 53910 T^{2} + 70 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 852 T + 388318 T^{2} + 852 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 668 T + 357854 T^{2} + 668 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 548 T + 478598 T^{2} + 548 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 284 T + 416037 T^{2} - 284 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 1162 T + 914766 T^{2} - 1162 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 806 T + 876422 T^{2} - 806 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 1510 T + 1282935 T^{2} + 1510 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 22 T + 648318 T^{2} + 22 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1540 T + 1446230 T^{2} + 1540 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 1323 T + p^{3} T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 472 T + 1378542 T^{2} - 472 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.824689642532781055412779886923, −9.585820233153366019684826410810, −9.017565630863225031261495599325, −8.650359298809845533792707796572, −8.180077243856625232575164993471, −8.055846566603943773123884705580, −6.81754186585719262347030634153, −6.79721207720674817166488250743, −6.56305373944618344180028384952, −6.09495961306927386497127579359, −5.44432443174751685863901085948, −4.77262482943253547564687470149, −4.03915261177024673816309329408, −3.82000147957200759201495419625, −3.23355933223705387078945104942, −2.99143539986641467788011415146, −1.62991969559195779040189106731, −1.39430900398765980754458917320, 0, 0,
1.39430900398765980754458917320, 1.62991969559195779040189106731, 2.99143539986641467788011415146, 3.23355933223705387078945104942, 3.82000147957200759201495419625, 4.03915261177024673816309329408, 4.77262482943253547564687470149, 5.44432443174751685863901085948, 6.09495961306927386497127579359, 6.56305373944618344180028384952, 6.79721207720674817166488250743, 6.81754186585719262347030634153, 8.055846566603943773123884705580, 8.180077243856625232575164993471, 8.650359298809845533792707796572, 9.017565630863225031261495599325, 9.585820233153366019684826410810, 9.824689642532781055412779886923
