| L(s) = 1 | − 2·3-s − 2·4-s − 5-s + 9-s + 4·12-s + 2·15-s + 2·20-s − 13·23-s − 4·25-s + 4·27-s − 10·31-s − 2·36-s − 6·37-s − 45-s + 7·47-s − 2·49-s + 14·53-s + 9·59-s − 4·60-s + 8·64-s + 10·67-s + 26·69-s − 3·71-s + 8·75-s − 11·81-s − 15·89-s + 26·92-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 4-s − 0.447·5-s + 1/3·9-s + 1.15·12-s + 0.516·15-s + 0.447·20-s − 2.71·23-s − 4/5·25-s + 0.769·27-s − 1.79·31-s − 1/3·36-s − 0.986·37-s − 0.149·45-s + 1.02·47-s − 2/7·49-s + 1.92·53-s + 1.17·59-s − 0.516·60-s + 64-s + 1.22·67-s + 3.13·69-s − 0.356·71-s + 0.923·75-s − 1.22·81-s − 1.58·89-s + 2.71·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + T + p T^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 52 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 104 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 53 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−7.04502838324138087262290742522, −6.65762051109697230207565543540, −6.18403517660781390542273208435, −5.65444849394794378272019115383, −5.39455959902491503973853144411, −5.25814444227930818243728717556, −4.37556314110803754461332293634, −4.12641506336196742819547078348, −3.87311776450695472561956997807, −3.34454435137950146335564399019, −2.33616056309322288369627632674, −2.02164582772337007198218623263, −1.05051773239065976697308404272, 0, 0,
1.05051773239065976697308404272, 2.02164582772337007198218623263, 2.33616056309322288369627632674, 3.34454435137950146335564399019, 3.87311776450695472561956997807, 4.12641506336196742819547078348, 4.37556314110803754461332293634, 5.25814444227930818243728717556, 5.39455959902491503973853144411, 5.65444849394794378272019115383, 6.18403517660781390542273208435, 6.65762051109697230207565543540, 7.04502838324138087262290742522
