| L(s) = 1 | − 2-s + 4-s − 8-s + 3·9-s − 2·13-s + 16-s − 8·17-s − 3·18-s − 10·25-s + 2·26-s + 8·29-s − 32-s + 8·34-s + 3·36-s + 14·37-s − 18·41-s + 49-s + 10·50-s − 2·52-s − 8·53-s − 8·58-s + 26·61-s + 64-s − 8·68-s − 3·72-s + 14·73-s − 14·74-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 9-s − 0.554·13-s + 1/4·16-s − 1.94·17-s − 0.707·18-s − 2·25-s + 0.392·26-s + 1.48·29-s − 0.176·32-s + 1.37·34-s + 1/2·36-s + 2.30·37-s − 2.81·41-s + 1/7·49-s + 1.41·50-s − 0.277·52-s − 1.09·53-s − 1.05·58-s + 3.32·61-s + 1/8·64-s − 0.970·68-s − 0.353·72-s + 1.63·73-s − 1.62·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 264992 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 264992 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.066385507\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.066385507\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
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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
−8.916781749804225553868026019180, −8.410332621243878174631771496694, −7.959201586617154759738546904094, −7.68283143195211494191877839909, −6.90516587374622685438056526369, −6.63299834857139013821510766042, −6.36335980553572410729518682566, −5.54975838409236888806497825912, −4.88738837750963504497893956492, −4.44236143050046711072138337998, −3.91571952847248384529393635172, −3.16464702872454555446785806554, −2.10681498002416439479097053291, −2.05625495312932408576461106462, −0.68078446997238380831438979083,
0.68078446997238380831438979083, 2.05625495312932408576461106462, 2.10681498002416439479097053291, 3.16464702872454555446785806554, 3.91571952847248384529393635172, 4.44236143050046711072138337998, 4.88738837750963504497893956492, 5.54975838409236888806497825912, 6.36335980553572410729518682566, 6.63299834857139013821510766042, 6.90516587374622685438056526369, 7.68283143195211494191877839909, 7.959201586617154759738546904094, 8.410332621243878174631771496694, 8.916781749804225553868026019180
