| L(s) = 1 | − 2-s + 4-s + 2·5-s − 8-s − 2·10-s + 16-s − 8·19-s + 2·20-s + 3·25-s − 12·29-s − 32-s + 8·38-s − 2·40-s + 16·43-s − 10·49-s − 3·50-s − 12·53-s + 12·58-s + 64-s − 8·67-s − 24·71-s − 20·73-s − 8·76-s + 2·80-s − 16·86-s − 16·95-s + 4·97-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.353·8-s − 0.632·10-s + 1/4·16-s − 1.83·19-s + 0.447·20-s + 3/5·25-s − 2.22·29-s − 0.176·32-s + 1.29·38-s − 0.316·40-s + 2.43·43-s − 1.42·49-s − 0.424·50-s − 1.64·53-s + 1.57·58-s + 1/8·64-s − 0.977·67-s − 2.84·71-s − 2.34·73-s − 0.917·76-s + 0.223·80-s − 1.72·86-s − 1.64·95-s + 0.406·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−8.808531037524113106929321069025, −8.330531568433955397282390142839, −7.73490121522356695217356839933, −7.33859557219653016096826874551, −6.86802385357238762255411937862, −6.21060412222094990512561243657, −5.84293355103774809066040280829, −5.62305662493911647485930324213, −4.48715934481021693344557314737, −4.39745787492882284514235740312, −3.37793372322877666574139691343, −2.75202257331423339947272598173, −1.98468742030680472449906296868, −1.53717311332283821140901478063, 0,
1.53717311332283821140901478063, 1.98468742030680472449906296868, 2.75202257331423339947272598173, 3.37793372322877666574139691343, 4.39745787492882284514235740312, 4.48715934481021693344557314737, 5.62305662493911647485930324213, 5.84293355103774809066040280829, 6.21060412222094990512561243657, 6.86802385357238762255411937862, 7.33859557219653016096826874551, 7.73490121522356695217356839933, 8.330531568433955397282390142839, 8.808531037524113106929321069025
