| L(s) = 1 | − 2-s + 4-s − 2·5-s − 8-s + 2·10-s + 16-s − 4·17-s − 2·20-s − 7·25-s − 8·29-s − 32-s + 4·34-s + 10·37-s + 2·40-s + 18·41-s + 49-s + 7·50-s − 24·53-s + 8·58-s + 64-s − 4·68-s − 4·73-s − 10·74-s − 2·80-s − 18·82-s + 8·85-s + 18·89-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 − 0.970·17-s − 0.447·20-s − 7/5·25-s − 1.48·29-s − 0.176·32-s + 0.685·34-s + 1.64·37-s + 0.316·40-s + 2.81·41-s + 1/7·49-s + 0.989·50-s − 3.29·53-s + 1.05·58-s + 1/8·64-s − 0.485·68-s − 0.468·73-s − 1.16·74-s − 0.223·80-s − 1.98·82-s + 0.867·85-s + 1.90·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143072 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143072 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.82180523610847379974298804156, −7.43396451799616319962951174764, −7.41731738918212721913056577334, −6.31189353147311880492962393494, −6.23227538486956081778076113568, −5.91137064624602427944931283698, −5.08207234913297047777508467306, −4.50854174640145518647490777803, −4.21595600702686973203131728090, −3.57313298834818934312909000630, −3.16207871637142803917205041928, −2.24049119020186278469176393363, −1.99414709782610576510843414106, −0.875062595129075836979928350629, 0,
0.875062595129075836979928350629, 1.99414709782610576510843414106, 2.24049119020186278469176393363, 3.16207871637142803917205041928, 3.57313298834818934312909000630, 4.21595600702686973203131728090, 4.50854174640145518647490777803, 5.08207234913297047777508467306, 5.91137064624602427944931283698, 6.23227538486956081778076113568, 6.31189353147311880492962393494, 7.41731738918212721913056577334, 7.43396451799616319962951174764, 7.82180523610847379974298804156
