| L(s) = 1 | + 4·3-s + 10·9-s + 20·27-s − 24·41-s − 16·43-s + 4·49-s + 16·67-s + 35·81-s − 24·89-s − 48·107-s + 44·121-s − 96·123-s + 127-s − 64·129-s + 131-s + 137-s + 139-s + 16·147-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 10/3·9-s + 3.84·27-s − 3.74·41-s − 2.43·43-s + 4/7·49-s + 1.95·67-s + 35/9·81-s − 2.54·89-s − 4.64·107-s + 4·121-s − 8.65·123-s + 0.0887·127-s − 5.63·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.31·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.569177150\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.569177150\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 94 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 + 94 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 190 T^{2} + p^{2} T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.84437107999515757029604054952, −5.48919479527498536386684897525, −5.47322983344339499908804413106, −5.09335812957088570680207045079, −4.95684601069507077375726667596, −4.85307968993678464529796509549, −4.82014774512490675407827899861, −4.31470727473056535141488689299, −4.20687837121113626552366424951, −3.82863118148668849192396788582, −3.82781902472090237279117887154, −3.57534614438738266602058295039, −3.45769240653537818604853541824, −3.26545991257441737057795762031, −3.00783085860334664774361951876, −2.66804575418411085169993260402, −2.60416910103091292378355961539, −2.34627932300160117978516015643, −2.23029950200667800324284693604, −1.62116629901489133674584384648, −1.57373969030464851546952209057, −1.49579243033920072884898908842, −1.31777832331629305834978479560, −0.50469519054015456670412738119, −0.36414243055218626506443927057,
0.36414243055218626506443927057, 0.50469519054015456670412738119, 1.31777832331629305834978479560, 1.49579243033920072884898908842, 1.57373969030464851546952209057, 1.62116629901489133674584384648, 2.23029950200667800324284693604, 2.34627932300160117978516015643, 2.60416910103091292378355961539, 2.66804575418411085169993260402, 3.00783085860334664774361951876, 3.26545991257441737057795762031, 3.45769240653537818604853541824, 3.57534614438738266602058295039, 3.82781902472090237279117887154, 3.82863118148668849192396788582, 4.20687837121113626552366424951, 4.31470727473056535141488689299, 4.82014774512490675407827899861, 4.85307968993678464529796509549, 4.95684601069507077375726667596, 5.09335812957088570680207045079, 5.47322983344339499908804413106, 5.48919479527498536386684897525, 5.84437107999515757029604054952
