| L(s) = 1 | − 2·4-s − 8·7-s + 16·13-s + 4·16-s − 32·19-s + 32·25-s + 16·28-s + 88·31-s − 68·37-s − 80·43-s − 50·49-s − 32·52-s + 100·61-s − 8·64-s + 16·67-s − 32·73-s + 64·76-s − 152·79-s − 128·91-s + 352·97-s − 64·100-s − 56·103-s + 112·109-s − 32·112-s − 46·121-s − 176·124-s + 127-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 8/7·7-s + 1.23·13-s + 1/4·16-s − 1.68·19-s + 1.27·25-s + 4/7·28-s + 2.83·31-s − 1.83·37-s − 1.86·43-s − 1.02·49-s − 0.615·52-s + 1.63·61-s − 1/8·64-s + 0.238·67-s − 0.438·73-s + 0.842·76-s − 1.92·79-s − 1.40·91-s + 3.62·97-s − 0.639·100-s − 0.543·103-s + 1.02·109-s − 2/7·112-s − 0.380·121-s − 1.41·124-s + 0.00787·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6215423735\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6215423735\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 32 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 14 T + p^{2} T^{2} )( 1 + 14 T + p^{2} T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 8 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 416 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 16 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 770 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 1664 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 44 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 34 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 1184 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 40 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 2782 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 4160 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 5810 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 50 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 7490 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 16 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 76 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 334 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 15680 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 176 T + p^{2} T^{2} )^{2} \) |
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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
−18.89347273255062280476635091455, −18.41989986415510726525675866418, −17.34631555644508276461662797034, −17.17386069467369261541271057371, −16.12940553476581195592109514968, −15.80362893975988818583436490781, −14.98901922992886842054569501792, −14.22146059509573790071482964497, −13.34283436675829660838174997309, −13.07300224456653051959814522774, −12.28179795767312081834728189015, −11.36515219725111670229259044791, −10.34383778911769437279554643238, −9.955810945311239495961961241930, −8.564573240899659569950682875282, −8.550251627536126227175555967214, −6.74703699913736062741911446047, −6.24803028236882973923729368575, −4.71803328341868101997909002331, −3.36175297440665047564543646315,
3.36175297440665047564543646315, 4.71803328341868101997909002331, 6.24803028236882973923729368575, 6.74703699913736062741911446047, 8.550251627536126227175555967214, 8.564573240899659569950682875282, 9.955810945311239495961961241930, 10.34383778911769437279554643238, 11.36515219725111670229259044791, 12.28179795767312081834728189015, 13.07300224456653051959814522774, 13.34283436675829660838174997309, 14.22146059509573790071482964497, 14.98901922992886842054569501792, 15.80362893975988818583436490781, 16.12940553476581195592109514968, 17.17386069467369261541271057371, 17.34631555644508276461662797034, 18.41989986415510726525675866418, 18.89347273255062280476635091455
