| L(s) = 1 | + 16·4-s − 324·11-s + 192·16-s + 624·23-s + 1.72e3·25-s − 2.72e3·29-s − 2.79e3·37-s − 632·43-s − 5.18e3·44-s − 2.07e3·53-s + 2.04e3·64-s − 2.92e4·67-s + 9.69e3·71-s − 7.94e3·79-s + 9.98e3·92-s + 2.76e4·100-s + 2.10e3·107-s + 4.00e3·109-s − 2.31e4·113-s − 4.35e4·116-s + 7.62e3·121-s + 127-s + 131-s + 137-s + 139-s − 4.46e4·148-s + 149-s + ⋯ |
| L(s) = 1 | + 4-s − 2.67·11-s + 3/4·16-s + 1.17·23-s + 2.76·25-s − 3.23·29-s − 2.03·37-s − 0.341·43-s − 2.67·44-s − 0.739·53-s + 1/2·64-s − 6.50·67-s + 1.92·71-s − 1.27·79-s + 1.17·92-s + 2.76·100-s + 0.183·107-s + 0.336·109-s − 1.81·113-s − 3.23·116-s + 0.520·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s − 2.03·148-s + 4.50e−5·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(3.591976051\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.591976051\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 1726 T^{2} + 1491171 T^{4} - 1726 p^{8} T^{6} + p^{16} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 162 T + 35555 T^{2} + 162 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 50980 T^{2} + 1849726854 T^{4} - 50980 p^{8} T^{6} + p^{16} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 940 p^{2} T^{2} + 383334 p^{4} T^{4} - 940 p^{10} T^{6} + p^{16} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 432940 T^{2} + 79003273254 T^{4} - 432940 p^{8} T^{6} + p^{16} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 312 T - 165070 T^{2} - 312 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 1362 T + 1874795 T^{2} + 1362 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 941806 T^{2} + 1059591694899 T^{4} - 941806 p^{8} T^{6} + p^{16} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 1396 T + 4217094 T^{2} + 1396 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 2317228 T^{2} + 5437495988070 T^{4} - 2317228 p^{8} T^{6} + p^{16} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 316 T - 3534234 T^{2} + 316 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 3578788 T^{2} - 6833831226042 T^{4} - 3578788 p^{8} T^{6} + p^{16} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 1038 T + 13256075 T^{2} + 1038 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 27657790 T^{2} + 481686647166339 T^{4} - 27657790 p^{8} T^{6} + p^{16} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 49777732 T^{2} + 997300062659526 T^{4} - 49777732 p^{8} T^{6} + p^{16} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 14600 T + 91702674 T^{2} + 14600 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 4848 T + 52412546 T^{2} - 4848 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 16586788 T^{2} + 661626455827398 T^{4} - 16586788 p^{8} T^{6} + p^{16} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 3974 T + 10279683 T^{2} + 3974 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 1452506 p T^{2} + 7047387074675283 T^{4} - 1452506 p^{9} T^{6} + p^{16} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 3460244 T^{2} - 5170773883613466 T^{4} + 3460244 p^{8} T^{6} + p^{16} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 347883310 T^{2} + 45930093674217747 T^{4} - 347883310 p^{8} T^{6} + p^{16} T^{8} \) |
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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
−6.63127122517899256033757818895, −6.47568303529374184746926331521, −6.26761334886667028840214639156, −6.07400520878187717453397285635, −5.54961620649477422415770001354, −5.43990931699374404851198113059, −5.33942902716893952291199111538, −5.13606954745598223907190740738, −4.99528498466812057821760324589, −4.69266564214631298985402755086, −4.25833522281845272503458479122, −3.95851010901310512558202141697, −3.92978791071410520279288609486, −3.21020562357548848973775870257, −3.10031177059807273507657361526, −2.92128188646139892775562600578, −2.78272318655122574892598861457, −2.72901191265182802405390759427, −1.89239657980900857657407972131, −1.86551390396101856228195060760, −1.62008134584695543201234750783, −1.43599821836200695973862841225, −0.75087166396158790246634180069, −0.35693736436136417517232119515, −0.32472216775137108153385741475,
0.32472216775137108153385741475, 0.35693736436136417517232119515, 0.75087166396158790246634180069, 1.43599821836200695973862841225, 1.62008134584695543201234750783, 1.86551390396101856228195060760, 1.89239657980900857657407972131, 2.72901191265182802405390759427, 2.78272318655122574892598861457, 2.92128188646139892775562600578, 3.10031177059807273507657361526, 3.21020562357548848973775870257, 3.92978791071410520279288609486, 3.95851010901310512558202141697, 4.25833522281845272503458479122, 4.69266564214631298985402755086, 4.99528498466812057821760324589, 5.13606954745598223907190740738, 5.33942902716893952291199111538, 5.43990931699374404851198113059, 5.54961620649477422415770001354, 6.07400520878187717453397285635, 6.26761334886667028840214639156, 6.47568303529374184746926331521, 6.63127122517899256033757818895
