| L(s) = 1 | + 4·3-s − 2·5-s + 6·9-s − 8·15-s − 6·17-s − 19-s − 7·25-s − 4·27-s + 8·31-s − 12·45-s − 5·49-s − 24·51-s − 4·57-s + 12·59-s − 26·61-s − 24·67-s + 4·71-s + 18·73-s − 28·75-s + 16·79-s − 37·81-s + 12·85-s + 32·93-s + 2·95-s − 20·101-s − 12·103-s + 4·107-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 0.894·5-s + 2·9-s − 2.06·15-s − 1.45·17-s − 0.229·19-s − 7/5·25-s − 0.769·27-s + 1.43·31-s − 1.78·45-s − 5/7·49-s − 3.36·51-s − 0.529·57-s + 1.56·59-s − 3.32·61-s − 2.93·67-s + 0.474·71-s + 2.10·73-s − 3.23·75-s + 1.80·79-s − 4.11·81-s + 1.30·85-s + 3.31·93-s + 0.205·95-s − 1.99·101-s − 1.18·103-s + 0.386·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 438976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 438976 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_1$ | \( 1 + T \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−8.392913880587579377462610956197, −7.980897390997220348818758273856, −7.62429229152375892340957919668, −7.43417964682464018577755229644, −6.44323514092267885981017350010, −6.36338832124191520338637131501, −5.50807847957788517516783104842, −4.74929106277827912586938097301, −4.15537276955970084853193543262, −3.93336710501963899561206806573, −3.26219872013992782989814019344, −2.82511261070703297705797963037, −2.27913891249710641141411097526, −1.70231193814897185426764814465, 0,
1.70231193814897185426764814465, 2.27913891249710641141411097526, 2.82511261070703297705797963037, 3.26219872013992782989814019344, 3.93336710501963899561206806573, 4.15537276955970084853193543262, 4.74929106277827912586938097301, 5.50807847957788517516783104842, 6.36338832124191520338637131501, 6.44323514092267885981017350010, 7.43417964682464018577755229644, 7.62429229152375892340957919668, 7.980897390997220348818758273856, 8.392913880587579377462610956197
