| L(s) = 1 | − 5-s + 3·9-s − 4·11-s − 4·13-s − 2·17-s − 4·19-s − 4·23-s − 4·29-s + 8·31-s − 6·37-s − 12·41-s − 16·43-s − 3·45-s − 4·47-s − 6·53-s + 4·55-s + 4·59-s + 2·61-s + 4·65-s − 8·67-s + 6·73-s − 32·83-s + 2·85-s + 6·89-s + 4·95-s − 28·97-s − 12·99-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 9-s − 1.20·11-s − 1.10·13-s − 0.485·17-s − 0.917·19-s − 0.834·23-s − 0.742·29-s + 1.43·31-s − 0.986·37-s − 1.87·41-s − 2.43·43-s − 0.447·45-s − 0.583·47-s − 0.824·53-s + 0.539·55-s + 0.520·59-s + 0.256·61-s + 0.496·65-s − 0.977·67-s + 0.702·73-s − 3.51·83-s + 0.216·85-s + 0.635·89-s + 0.410·95-s − 2.84·97-s − 1.20·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3841600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3841600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 6 T - T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 4 T - 31 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−8.696351068262811813343737346114, −8.625490381856910231652699747815, −8.098616550034629606758454378740, −7.996911434481416168989320011577, −7.31981883216940646131873622496, −7.10087484057734672004397554693, −6.58473123535654964919161987560, −6.50754188944160612767494695599, −5.60895057561868352633980109703, −5.36987291517909926555659346102, −4.69720426901233848039597566473, −4.68485112461023760527233830364, −4.08278301190477344275218359540, −3.58370742225812749430113189290, −2.98791496680025517636856001720, −2.56543046560386317481587601774, −1.82805362826891588622599990680, −1.55878785509549504847810268203, 0, 0,
1.55878785509549504847810268203, 1.82805362826891588622599990680, 2.56543046560386317481587601774, 2.98791496680025517636856001720, 3.58370742225812749430113189290, 4.08278301190477344275218359540, 4.68485112461023760527233830364, 4.69720426901233848039597566473, 5.36987291517909926555659346102, 5.60895057561868352633980109703, 6.50754188944160612767494695599, 6.58473123535654964919161987560, 7.10087484057734672004397554693, 7.31981883216940646131873622496, 7.996911434481416168989320011577, 8.098616550034629606758454378740, 8.625490381856910231652699747815, 8.696351068262811813343737346114
