| L(s) = 1 | − 2·3-s + 5-s − 7-s + 3·9-s − 7·11-s − 2·13-s − 2·15-s + 17-s − 2·19-s + 2·21-s − 10·23-s − 5·25-s − 4·27-s + 4·31-s + 14·33-s − 35-s − 16·37-s + 4·39-s − 2·41-s − 43-s + 3·45-s + 47-s − 9·49-s − 2·51-s − 8·53-s − 7·55-s + 4·57-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s − 0.377·7-s + 9-s − 2.11·11-s − 0.554·13-s − 0.516·15-s + 0.242·17-s − 0.458·19-s + 0.436·21-s − 2.08·23-s − 25-s − 0.769·27-s + 0.718·31-s + 2.43·33-s − 0.169·35-s − 2.63·37-s + 0.640·39-s − 0.312·41-s − 0.152·43-s + 0.447·45-s + 0.145·47-s − 9/7·49-s − 0.280·51-s − 1.09·53-s − 0.943·55-s + 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T + 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 7 T + 30 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_4$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 10 T + 54 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 2 T + 66 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + T + 82 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - T + 56 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - T + 84 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 5 T - 56 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 142 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−10.04861278183804889970414996523, −9.788453288912534744179042137662, −9.150349289545777322266521821839, −8.598341296363830787311762210788, −7.946528061505756668603912700929, −7.83317291651819864639250492514, −7.44053253339425069190783619267, −6.63714480820845956926603716281, −6.46444898808657762691665500108, −5.90083704059303951170111712183, −5.45272936977689125253089538386, −5.27661681924766736478509508329, −4.57008670893721800385386637446, −4.29891914253423664511325975669, −3.34597243996746572161592963996, −2.96494741148044988785195310032, −1.99601286959312429768433591503, −1.77028391308079892196811307389, 0, 0,
1.77028391308079892196811307389, 1.99601286959312429768433591503, 2.96494741148044988785195310032, 3.34597243996746572161592963996, 4.29891914253423664511325975669, 4.57008670893721800385386637446, 5.27661681924766736478509508329, 5.45272936977689125253089538386, 5.90083704059303951170111712183, 6.46444898808657762691665500108, 6.63714480820845956926603716281, 7.44053253339425069190783619267, 7.83317291651819864639250492514, 7.946528061505756668603912700929, 8.598341296363830787311762210788, 9.150349289545777322266521821839, 9.788453288912534744179042137662, 10.04861278183804889970414996523
