| L(s) = 1 | − 2·5-s + 4·7-s + 2·11-s + 8·19-s + 3·25-s + 4·29-s + 4·31-s − 8·35-s + 4·37-s + 4·41-s + 4·43-s − 8·47-s + 4·53-s − 4·55-s − 12·59-s + 12·61-s − 4·71-s + 8·77-s + 16·79-s − 12·83-s + 4·89-s − 16·95-s + 12·97-s + 20·101-s − 12·107-s − 4·109-s + 4·113-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.51·7-s + 0.603·11-s + 1.83·19-s + 3/5·25-s + 0.742·29-s + 0.718·31-s − 1.35·35-s + 0.657·37-s + 0.624·41-s + 0.609·43-s − 1.16·47-s + 0.549·53-s − 0.539·55-s − 1.56·59-s + 1.53·61-s − 0.474·71-s + 0.911·77-s + 1.80·79-s − 1.31·83-s + 0.423·89-s − 1.64·95-s + 1.21·97-s + 1.99·101-s − 1.16·107-s − 0.383·109-s + 0.376·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62726400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62726400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.712273754\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.712273754\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - 4 T + 16 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 58 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 40 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 12 T + 146 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 74 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 96 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 184 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 12 T + 222 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
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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
−7.915267519737862949108166486253, −7.84196533970187846806394745874, −7.27455231148917980663780008306, −7.22706338004649759920767944024, −6.62579518065423300431740768651, −6.39122541649336575227585235504, −5.86797783638690144080705637099, −5.57556387930043359208140976744, −4.98992372658770539455532675687, −4.93155828520612439575850495942, −4.39770839082835835666110504710, −4.33628768835155389199481604448, −3.59228961202000058700130130778, −3.47615209050170800935456851052, −2.83488744193346577323784261823, −2.63328086337728497804891513776, −1.72935820272930836841945637350, −1.64536728341114238413282952998, −0.77476658773066623187586469196, −0.75366034043447184966304522189,
0.75366034043447184966304522189, 0.77476658773066623187586469196, 1.64536728341114238413282952998, 1.72935820272930836841945637350, 2.63328086337728497804891513776, 2.83488744193346577323784261823, 3.47615209050170800935456851052, 3.59228961202000058700130130778, 4.33628768835155389199481604448, 4.39770839082835835666110504710, 4.93155828520612439575850495942, 4.98992372658770539455532675687, 5.57556387930043359208140976744, 5.86797783638690144080705637099, 6.39122541649336575227585235504, 6.62579518065423300431740768651, 7.22706338004649759920767944024, 7.27455231148917980663780008306, 7.84196533970187846806394745874, 7.915267519737862949108166486253
