| L(s) = 1 | − 12·3-s − 54·9-s + 536·11-s + 380·25-s + 2.05e3·27-s − 6.43e3·33-s − 6.72e3·49-s − 5.52e3·59-s + 1.95e4·73-s − 4.56e3·75-s − 8.82e3·81-s − 2.36e4·83-s + 2.34e4·97-s − 2.89e4·99-s + 8.50e4·107-s + 1.20e5·121-s + 127-s + 131-s + 137-s + 139-s + 8.06e4·147-s + 149-s + 151-s + 157-s + 163-s + 167-s − 9.31e4·169-s + ⋯ |
| L(s) = 1 | − 4/3·3-s − 2/3·9-s + 4.42·11-s + 0.607·25-s + 2.81·27-s − 5.90·33-s − 2.80·49-s − 1.58·59-s + 3.67·73-s − 0.810·75-s − 1.34·81-s − 3.43·83-s + 2.49·97-s − 2.95·99-s + 7.43·107-s + 8.26·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 3.73·147-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 − 3.26·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(4.121363472\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.121363472\) |
| \(L(3)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 + 2 p T + p^{4} T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 - 38 p T^{2} + p^{8} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 3362 T^{2} + p^{8} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 134 T + p^{4} T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 + 46558 T^{2} + p^{8} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 110594 T^{2} + p^{8} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 18434 T^{2} + p^{8} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 144962 T^{2} + p^{8} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 779522 T^{2} + p^{8} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 636002 T^{2} + p^{8} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 1156322 T^{2} + p^{8} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 3988034 T^{2} + p^{8} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 6657602 T^{2} + p^{8} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 3123842 T^{2} + p^{8} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 13118402 T^{2} + p^{8} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 1382 T + p^{4} T^{2} )^{4} \) |
| 61 | $C_2^2$ | \( ( 1 - 27588002 T^{2} + p^{8} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 40267394 T^{2} + p^{8} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 47090882 T^{2} + p^{8} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4894 T + p^{4} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 75709922 T^{2} + p^{8} T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 5914 T + p^{4} T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 - 146 p T + p^{4} T^{2} )^{2}( 1 + 146 p T + p^{4} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 5858 T + p^{4} T^{2} )^{4} \) |
| 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
−7.42971798895615412475464463932, −7.16704250306212038308531364638, −6.94036016089278765614015042094, −6.58591718895229054336555355856, −6.55683486557240778684909366046, −6.24908175476181567170742756798, −6.01132966089653162991311009247, −6.00781262385241006909935091129, −5.75443877622999579663147582544, −5.07849028101878372972399635501, −4.89334956767050092290594417300, −4.83394479051005586261730082593, −4.37086732226604711774496694337, −4.27962885669750693861051280121, −3.64394360595061051881950356873, −3.55162809193684666359752005690, −3.32660931717580444725000056803, −3.05906629281221304184114089871, −2.48426995633550173275920915321, −1.88333305217620891750171785459, −1.76862549458342717227578103905, −1.30131357889534417322863241613, −0.985631929268675414358453245315, −0.61970365470539023692694636437, −0.39022030855187228989397258222,
0.39022030855187228989397258222, 0.61970365470539023692694636437, 0.985631929268675414358453245315, 1.30131357889534417322863241613, 1.76862549458342717227578103905, 1.88333305217620891750171785459, 2.48426995633550173275920915321, 3.05906629281221304184114089871, 3.32660931717580444725000056803, 3.55162809193684666359752005690, 3.64394360595061051881950356873, 4.27962885669750693861051280121, 4.37086732226604711774496694337, 4.83394479051005586261730082593, 4.89334956767050092290594417300, 5.07849028101878372972399635501, 5.75443877622999579663147582544, 6.00781262385241006909935091129, 6.01132966089653162991311009247, 6.24908175476181567170742756798, 6.55683486557240778684909366046, 6.58591718895229054336555355856, 6.94036016089278765614015042094, 7.16704250306212038308531364638, 7.42971798895615412475464463932
