| L(s) = 1 | − 432·11-s + 412·16-s − 2.91e3·31-s + 6.04e3·41-s + 1.83e4·61-s + 1.72e3·71-s − 1.27e3·81-s + 1.36e4·101-s + 5.80e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 1.77e5·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 3.57·11-s + 1.60·16-s − 3.03·31-s + 3.59·41-s + 4.93·61-s + 0.342·71-s − 0.194·81-s + 1.33·101-s + 3.96·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + 3.34e−5·173-s − 5.74·176-s + 3.12e−5·179-s + 3.05e−5·181-s + 2.74e−5·191-s + 2.68e−5·193-s + 2.57e−5·197-s + 2.52e−5·199-s + 2.24e−5·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390625 ^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.247321468\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.247321468\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 - 103 p^{2} T^{4} + p^{16} T^{8} \) |
| 3 | $C_2^3$ | \( 1 + 142 p^{2} T^{4} + p^{16} T^{8} \) |
| 7 | $C_2^3$ | \( 1 - 195298 p^{2} T^{4} + p^{16} T^{8} \) |
| 11 | $C_2$ | \( ( 1 + 108 T + p^{4} T^{2} )^{4} \) |
| 13 | $C_2^3$ | \( 1 - 1624225342 T^{4} + p^{16} T^{8} \) |
| 17 | $C_2^3$ | \( 1 + 13727462018 T^{4} + p^{16} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 241042 T^{2} + p^{8} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 68080016962 T^{4} + p^{16} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 758462 T^{2} + p^{8} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 728 T + p^{4} T^{2} )^{4} \) |
| 37 | $C_2^3$ | \( 1 + 1254529400258 T^{4} + p^{16} T^{8} \) |
| 41 | $C_2$ | \( ( 1 - 1512 T + p^{4} T^{2} )^{4} \) |
| 43 | $C_2^3$ | \( 1 + 23329889084798 T^{4} + p^{16} T^{8} \) |
| 47 | $C_2^3$ | \( 1 + 46379759106878 T^{4} + p^{16} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 112877432101822 T^{4} + p^{16} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 9946322 T^{2} + p^{8} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4592 T + p^{4} T^{2} )^{4} \) |
| 67 | $C_2^3$ | \( 1 - 488793100954882 T^{4} + p^{16} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - 432 T + p^{4} T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 - 937201474520062 T^{4} + p^{16} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 245438 T^{2} + p^{8} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 4278306619448318 T^{4} + p^{16} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 49548418 T^{2} + p^{8} T^{4} )^{2} \) |
| 97 | $C_2^3$ | \( 1 - 13524937897425022 T^{4} + p^{16} T^{8} \) |
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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
−12.78192932180774845523898685605, −12.53624717842331331620529175073, −11.67478839897284912867405172988, −11.41303455750850584607831408048, −11.05307183057885953789274121068, −10.56098813111916343885663426486, −10.48227733821222530044157346234, −10.22638633811206331702119818654, −9.696939734558382659396119093107, −9.264688062490167833544554826785, −8.929968485832677490939106734204, −8.114828441273900496973493066252, −7.934702921357344420241828112325, −7.921994448892469209051337068858, −7.14420310823011059125067556104, −7.11064104046463757865937800195, −5.99616693891148073895995086723, −5.54576003607194353984595342108, −5.45450962887272460739985452436, −5.04571466633359189653701663886, −4.08650463379581505556077073696, −3.51054686644711393196966321495, −2.66521036695721992850857802600, −2.23756114453069697122663772602, −0.58416573543864551858608430798,
0.58416573543864551858608430798, 2.23756114453069697122663772602, 2.66521036695721992850857802600, 3.51054686644711393196966321495, 4.08650463379581505556077073696, 5.04571466633359189653701663886, 5.45450962887272460739985452436, 5.54576003607194353984595342108, 5.99616693891148073895995086723, 7.11064104046463757865937800195, 7.14420310823011059125067556104, 7.921994448892469209051337068858, 7.934702921357344420241828112325, 8.114828441273900496973493066252, 8.929968485832677490939106734204, 9.264688062490167833544554826785, 9.696939734558382659396119093107, 10.22638633811206331702119818654, 10.48227733821222530044157346234, 10.56098813111916343885663426486, 11.05307183057885953789274121068, 11.41303455750850584607831408048, 11.67478839897284912867405172988, 12.53624717842331331620529175073, 12.78192932180774845523898685605
