| L(s) = 1 | + 2·3-s − 2·5-s − 9-s − 2·11-s − 4·15-s − 12·17-s + 4·19-s − 14·23-s + 25-s − 6·27-s − 16·29-s − 2·31-s − 4·33-s + 6·37-s + 4·43-s + 2·45-s − 8·47-s − 12·49-s − 24·51-s + 4·55-s + 8·57-s + 2·59-s + 6·67-s − 28·69-s − 2·71-s − 16·73-s + 2·75-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s − 1/3·9-s − 0.603·11-s − 1.03·15-s − 2.91·17-s + 0.917·19-s − 2.91·23-s + 1/5·25-s − 1.15·27-s − 2.97·29-s − 0.359·31-s − 0.696·33-s + 0.986·37-s + 0.609·43-s + 0.298·45-s − 1.16·47-s − 1.71·49-s − 3.36·51-s + 0.539·55-s + 1.05·57-s + 0.260·59-s + 0.733·67-s − 3.37·69-s − 0.237·71-s − 1.87·73-s + 0.230·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1982464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1982464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 12 T + 4 p T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 40 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 14 T + 93 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 6 T + 75 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 82 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 2 T - 43 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 6 T - 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 2 T + 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 16 T + 160 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 4 T + 154 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 28 T + 354 T^{2} - 28 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 2 T + 171 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 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
−9.307792892000514667433720601060, −8.941278534885239021127729197559, −8.323259246311900458736051288468, −8.317775442759805899000068256204, −7.74507549059206434494460567708, −7.62390622147860683081293782265, −7.06866181070146480199960466027, −6.55334978843337743957316088391, −5.96557144008317646980639787980, −5.77572163956453322882330457866, −5.09578898975605732472636407584, −4.47180994629019935382320498234, −4.14875096834245229596544178879, −3.67772654398545225543015562536, −3.30253751061432357148054251435, −2.62960626220228083891303419450, −1.98358057460718470239393544702, −1.96833375484122923003211673218, 0, 0,
1.96833375484122923003211673218, 1.98358057460718470239393544702, 2.62960626220228083891303419450, 3.30253751061432357148054251435, 3.67772654398545225543015562536, 4.14875096834245229596544178879, 4.47180994629019935382320498234, 5.09578898975605732472636407584, 5.77572163956453322882330457866, 5.96557144008317646980639787980, 6.55334978843337743957316088391, 7.06866181070146480199960466027, 7.62390622147860683081293782265, 7.74507549059206434494460567708, 8.317775442759805899000068256204, 8.323259246311900458736051288468, 8.941278534885239021127729197559, 9.307792892000514667433720601060
