| L(s) = 1 | + 3·5-s − 7-s − 3·9-s − 3·11-s − 13-s + 12·17-s − 8·19-s − 3·23-s + 5·25-s + 3·29-s + 5·31-s − 3·35-s − 4·37-s − 3·41-s + 43-s − 9·45-s − 9·47-s + 7·49-s + 12·53-s − 9·55-s + 3·59-s − 13·61-s + 3·63-s − 3·65-s + 7·67-s + 24·71-s − 20·73-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 0.377·7-s − 9-s − 0.904·11-s − 0.277·13-s + 2.91·17-s − 1.83·19-s − 0.625·23-s + 25-s + 0.557·29-s + 0.898·31-s − 0.507·35-s − 0.657·37-s − 0.468·41-s + 0.152·43-s − 1.34·45-s − 1.31·47-s + 49-s + 1.64·53-s − 1.21·55-s + 0.390·59-s − 1.66·61-s + 0.377·63-s − 0.372·65-s + 0.855·67-s + 2.84·71-s − 2.34·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.865463382\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.865463382\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 9 T + 34 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 11 T + 42 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 11 T + 24 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
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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
−10.54316616454141044819462304066, −10.35062920676357758302790618139, −10.31096860861319511967388779965, −9.629299710812925128844189685190, −9.410188143575859190711307273622, −8.566916073724740167392442946972, −8.447248011391483894540889885781, −7.899711178837467910326372702383, −7.49006187641192219307333481718, −6.69694017851654113679538556225, −6.34856182563885755102335982448, −5.77046107631856245557002806581, −5.60836974882995500954408250760, −5.07113302808418267133773026323, −4.50293706671950974106840890426, −3.46573541860135892652635766635, −3.20140224840601163360589573843, −2.38424784963553963887194806924, −1.99699374638027380507059778361, −0.77188997283383854367890158644,
0.77188997283383854367890158644, 1.99699374638027380507059778361, 2.38424784963553963887194806924, 3.20140224840601163360589573843, 3.46573541860135892652635766635, 4.50293706671950974106840890426, 5.07113302808418267133773026323, 5.60836974882995500954408250760, 5.77046107631856245557002806581, 6.34856182563885755102335982448, 6.69694017851654113679538556225, 7.49006187641192219307333481718, 7.899711178837467910326372702383, 8.447248011391483894540889885781, 8.566916073724740167392442946972, 9.410188143575859190711307273622, 9.629299710812925128844189685190, 10.31096860861319511967388779965, 10.35062920676357758302790618139, 10.54316616454141044819462304066
