| L(s) = 1 | − 2·2-s − 2·3-s + 3·4-s + 5-s + 4·6-s − 2·7-s − 4·8-s + 3·9-s − 2·10-s − 5·11-s − 6·12-s + 4·14-s − 2·15-s + 5·16-s + 17-s − 6·18-s − 5·19-s + 3·20-s + 4·21-s + 10·22-s + 3·23-s + 8·24-s + 25-s − 4·27-s − 6·28-s − 29-s + 4·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.447·5-s + 1.63·6-s − 0.755·7-s − 1.41·8-s + 9-s − 0.632·10-s − 1.50·11-s − 1.73·12-s + 1.06·14-s − 0.516·15-s + 5/4·16-s + 0.242·17-s − 1.41·18-s − 1.14·19-s + 0.670·20-s + 0.872·21-s + 2.13·22-s + 0.625·23-s + 1.63·24-s + 1/5·25-s − 0.769·27-s − 1.13·28-s − 0.185·29-s + 0.730·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50381604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50381604 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 - T - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 5 T + 18 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T + 24 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 5 T + 34 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 3 T + 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + T + 48 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 15 T + 120 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + T - 6 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 - 4 T - 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 9 T + 132 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2 T + 94 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 9 T + 156 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 6 T + 126 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 14 T + 174 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−7.62133593856147588864974082567, −7.57976194336651102367932624854, −6.83229182550392748881287348510, −6.79553145719867227607933891596, −6.63963338988380843288107994114, −6.08719747822843918064729502805, −5.63247540109401212554825950848, −5.44779964744874416962042712709, −5.15011245663189714527164051163, −4.79799817969987356744483761555, −3.96679014257129331235462057862, −3.83402339354698531677533686262, −3.27077128899533600048698914975, −2.62695951803637829494216707010, −2.30762536580100188551238448618, −2.11247323564436704233612143946, −1.22100104579519576297249934945, −0.914114483621225028889625908786, 0, 0,
0.914114483621225028889625908786, 1.22100104579519576297249934945, 2.11247323564436704233612143946, 2.30762536580100188551238448618, 2.62695951803637829494216707010, 3.27077128899533600048698914975, 3.83402339354698531677533686262, 3.96679014257129331235462057862, 4.79799817969987356744483761555, 5.15011245663189714527164051163, 5.44779964744874416962042712709, 5.63247540109401212554825950848, 6.08719747822843918064729502805, 6.63963338988380843288107994114, 6.79553145719867227607933891596, 6.83229182550392748881287348510, 7.57976194336651102367932624854, 7.62133593856147588864974082567
