| L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s + 5·16-s − 2·17-s + 2·19-s + 2·23-s + 6·32-s − 4·34-s + 4·38-s + 4·46-s − 2·47-s + 2·61-s + 7·64-s − 6·68-s + 6·76-s − 4·83-s + 6·92-s − 4·94-s − 4·107-s − 2·109-s + 2·113-s + 4·122-s + 127-s + 8·128-s + 131-s − 8·136-s + ⋯ |
| L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s + 5·16-s − 2·17-s + 2·19-s + 2·23-s + 6·32-s − 4·34-s + 4·38-s + 4·46-s − 2·47-s + 2·61-s + 7·64-s − 6·68-s + 6·76-s − 4·83-s + 6·92-s − 4·94-s − 4·107-s − 2·109-s + 2·113-s + 4·122-s + 127-s + 8·128-s + 131-s − 8·136-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(6.552958166\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.552958166\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + T^{4} \) |
| 11 | $C_2^2$ | \( 1 + T^{4} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + T^{4} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + T^{4} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2^2$ | \( 1 + T^{4} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 83 | $C_1$ | \( ( 1 + T )^{4} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + T^{4} \) |
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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
−8.790992238917499576246919786578, −8.478989919133056169283558066796, −8.026537898858517693773409798944, −7.57513745641619228211733496290, −7.18856308503068707964419483434, −6.93020166853298988510859527003, −6.55904275422440354084720007923, −6.47241231574239751317512153207, −5.63142473548678773609038152371, −5.50470912544726460002159708056, −4.97765505645303978347063780143, −4.94797969430622192602973608200, −4.20836516324813963644486868443, −4.11156005976907006916252028381, −3.40537873563978453697267780075, −3.13808068165915260415013603914, −2.58104332158666329411768478999, −2.43928531236859538920031795290, −1.44904393535381829465750433600, −1.28524364473144289524865885834,
1.28524364473144289524865885834, 1.44904393535381829465750433600, 2.43928531236859538920031795290, 2.58104332158666329411768478999, 3.13808068165915260415013603914, 3.40537873563978453697267780075, 4.11156005976907006916252028381, 4.20836516324813963644486868443, 4.94797969430622192602973608200, 4.97765505645303978347063780143, 5.50470912544726460002159708056, 5.63142473548678773609038152371, 6.47241231574239751317512153207, 6.55904275422440354084720007923, 6.93020166853298988510859527003, 7.18856308503068707964419483434, 7.57513745641619228211733496290, 8.026537898858517693773409798944, 8.478989919133056169283558066796, 8.790992238917499576246919786578
