| L(s) = 1 | + 8·5-s + 2·7-s − 11·9-s + 28·11-s + 46·17-s + 20·19-s + 2·23-s − 2·25-s + 16·35-s + 136·43-s − 88·45-s − 52·47-s − 95·49-s + 224·55-s + 80·61-s − 22·63-s − 14·73-s + 56·77-s + 40·81-s + 64·83-s + 368·85-s + 160·95-s − 308·99-s − 28·101-s + 16·115-s + 92·119-s + 346·121-s + ⋯ |
| L(s) = 1 | + 8/5·5-s + 2/7·7-s − 1.22·9-s + 2.54·11-s + 2.70·17-s + 1.05·19-s + 2/23·23-s − 0.0799·25-s + 0.457·35-s + 3.16·43-s − 1.95·45-s − 1.10·47-s − 1.93·49-s + 4.07·55-s + 1.31·61-s − 0.349·63-s − 0.191·73-s + 8/11·77-s + 0.493·81-s + 0.771·83-s + 4.32·85-s + 1.68·95-s − 3.11·99-s − 0.277·101-s + 0.139·115-s + 0.773·119-s + 2.85·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(6.102580181\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.102580181\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 - 20 T + p^{2} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 11 T^{2} + p^{4} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 14 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 77 T^{2} + p^{4} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 23 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + p^{2} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 23 p T^{2} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 878 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 1694 T^{2} + p^{4} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 2318 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 68 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 26 T + p^{2} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 907 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6701 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 40 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 8717 T^{2} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 9038 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 3086 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 32 T + p^{2} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 862 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 9422 T^{2} + p^{4} T^{4} \) |
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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
−9.849767427588763911175565021905, −9.322302678789353060981181437082, −9.260830292487217014082670146261, −8.522736283913833552330693976924, −8.249045386689032405343335985171, −7.54129496404459309833062785879, −7.47590301381111272033370804583, −6.65462470832823713973030884777, −6.24332934296524810959101456604, −5.98887171304563702153917503313, −5.57991657693264588891736896762, −5.35349135037206143523153771266, −4.71885420762624143477846122402, −3.90215098099036191626260203998, −3.61064707848301841295365696928, −3.08577152874536546932261570591, −2.48785799172814581395880074537, −1.70870053072043805899752136022, −1.30987901373179710315103427509, −0.818038274152329333782654862143,
0.818038274152329333782654862143, 1.30987901373179710315103427509, 1.70870053072043805899752136022, 2.48785799172814581395880074537, 3.08577152874536546932261570591, 3.61064707848301841295365696928, 3.90215098099036191626260203998, 4.71885420762624143477846122402, 5.35349135037206143523153771266, 5.57991657693264588891736896762, 5.98887171304563702153917503313, 6.24332934296524810959101456604, 6.65462470832823713973030884777, 7.47590301381111272033370804583, 7.54129496404459309833062785879, 8.249045386689032405343335985171, 8.522736283913833552330693976924, 9.260830292487217014082670146261, 9.322302678789353060981181437082, 9.849767427588763911175565021905
