| L(s) = 1 | + 2·2-s + 3·4-s + 2·5-s + 4·7-s + 4·8-s + 4·10-s + 4·13-s + 8·14-s + 5·16-s − 2·19-s + 6·20-s − 6·23-s + 3·25-s + 8·26-s + 12·28-s + 4·31-s + 6·32-s + 8·35-s + 16·37-s − 4·38-s + 8·40-s + 4·43-s − 12·46-s − 6·47-s + 49-s + 6·50-s + 12·52-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 0.894·5-s + 1.51·7-s + 1.41·8-s + 1.26·10-s + 1.10·13-s + 2.13·14-s + 5/4·16-s − 0.458·19-s + 1.34·20-s − 1.25·23-s + 3/5·25-s + 1.56·26-s + 2.26·28-s + 0.718·31-s + 1.06·32-s + 1.35·35-s + 2.63·37-s − 0.648·38-s + 1.26·40-s + 0.609·43-s − 1.76·46-s − 0.875·47-s + 1/7·49-s + 0.848·50-s + 1.66·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.870226003\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.870226003\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - 4 T + 15 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T - 9 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 43 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 91 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 18 T + 175 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 43 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 24 T + 298 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 16 T + 210 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
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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
−10.69013747202860526075941210787, −10.21969151683091560217746778590, −9.527707309423587890543651227738, −9.459756714675809609257892207026, −8.566047229404752750775180094859, −8.153278929527552134932684103769, −7.953045790428211410538060394308, −7.52025869314965325002620741571, −6.54018877143271356321352043054, −6.53670409643800207145069950924, −5.91395026966841704825980929023, −5.70888943723484864337617129638, −4.90678652249640476417375257666, −4.80530670438894117817852214319, −4.01469649409344327088736069597, −3.93320109737053838852390544097, −2.75092304501132463516783791043, −2.61126978492149657347030631221, −1.54178879206670646241964012180, −1.45394014886211029209340762042,
1.45394014886211029209340762042, 1.54178879206670646241964012180, 2.61126978492149657347030631221, 2.75092304501132463516783791043, 3.93320109737053838852390544097, 4.01469649409344327088736069597, 4.80530670438894117817852214319, 4.90678652249640476417375257666, 5.70888943723484864337617129638, 5.91395026966841704825980929023, 6.53670409643800207145069950924, 6.54018877143271356321352043054, 7.52025869314965325002620741571, 7.953045790428211410538060394308, 8.153278929527552134932684103769, 8.566047229404752750775180094859, 9.459756714675809609257892207026, 9.527707309423587890543651227738, 10.21969151683091560217746778590, 10.69013747202860526075941210787
