| L(s) = 1 | − 4·3-s + 4·9-s + 4·27-s + 24·31-s + 24·37-s + 24·41-s − 20·43-s + 4·49-s + 20·67-s − 24·71-s − 48·79-s − 10·81-s − 12·83-s + 24·89-s − 96·93-s − 60·107-s − 96·111-s + 20·121-s − 96·123-s + 127-s + 80·129-s + 131-s + 137-s + 139-s − 16·147-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 4/3·9-s + 0.769·27-s + 4.31·31-s + 3.94·37-s + 3.74·41-s − 3.04·43-s + 4/7·49-s + 2.44·67-s − 2.84·71-s − 5.40·79-s − 1.11·81-s − 1.31·83-s + 2.54·89-s − 9.95·93-s − 5.80·107-s − 9.11·111-s + 1.81·121-s − 8.65·123-s + 0.0887·127-s + 7.04·129-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7372505403\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7372505403\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( ( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 4 T^{2} - 6 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 186 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $D_{4}$ | \( ( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 41 | $D_{4}$ | \( ( 1 - 12 T + 106 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 10 T + 84 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 116 T^{2} + 6810 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 1974 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 10 T + 132 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 12 T + 70 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 4230 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) |
| 83 | $D_{4}$ | \( ( 1 + 6 T + 172 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 16806 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.54022699345926043927241430258, −6.30005149960429090287933652761, −6.25617107940457083617485891599, −6.05121611748413660395772210668, −5.83847237913057077148854493682, −5.76348645714371591265766943086, −5.46570457749523732557635948352, −5.31790890324813998998751405188, −5.04358314280262983916860828749, −4.74034609792205732734381498466, −4.36724675804943809310919661332, −4.35285653583575189449423578424, −4.29131285372188037807306536014, −4.13495289458410167344145611527, −3.56660631563327664133064224783, −3.10284661348298030400835026191, −2.94596198990756254493241844612, −2.70340260420984196844832641315, −2.51579777041534943932077382233, −2.42643109193513079283957715870, −1.65847103499722340426603297638, −1.28010753681688952047939959544, −1.01889646950838046798308843526, −0.72655362401117505216545844208, −0.28313718598906060654198519889,
0.28313718598906060654198519889, 0.72655362401117505216545844208, 1.01889646950838046798308843526, 1.28010753681688952047939959544, 1.65847103499722340426603297638, 2.42643109193513079283957715870, 2.51579777041534943932077382233, 2.70340260420984196844832641315, 2.94596198990756254493241844612, 3.10284661348298030400835026191, 3.56660631563327664133064224783, 4.13495289458410167344145611527, 4.29131285372188037807306536014, 4.35285653583575189449423578424, 4.36724675804943809310919661332, 4.74034609792205732734381498466, 5.04358314280262983916860828749, 5.31790890324813998998751405188, 5.46570457749523732557635948352, 5.76348645714371591265766943086, 5.83847237913057077148854493682, 6.05121611748413660395772210668, 6.25617107940457083617485891599, 6.30005149960429090287933652761, 6.54022699345926043927241430258
