L(s) = 1 | − 2·3-s + 2·9-s + 16-s − 8·23-s + 25-s − 2·27-s + 2·37-s − 2·47-s − 2·48-s + 2·53-s + 8·67-s + 16·69-s − 4·71-s − 2·75-s + 81-s + 2·97-s + 2·103-s − 4·111-s + 2·113-s + 127-s + 131-s + 137-s + 139-s + 4·141-s + 2·144-s + 149-s + 151-s + ⋯ |
L(s) = 1 | − 2·3-s + 2·9-s + 16-s − 8·23-s + 25-s − 2·27-s + 2·37-s − 2·47-s − 2·48-s + 2·53-s + 8·67-s + 16·69-s − 4·71-s − 2·75-s + 81-s + 2·97-s + 2·103-s − 4·111-s + 2·113-s + 127-s + 131-s + 137-s + 139-s + 4·141-s + 2·144-s + 149-s + 151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1444532389\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1444532389\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \) |
| 3 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \) |
| 7 | \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \) |
| 13 | \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \) |
| 17 | \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \) |
| 19 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 23 | \( ( 1 + T )^{8}( 1 + T^{2} )^{4} \) |
| 29 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 31 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 37 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \) |
| 41 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 43 | \( ( 1 + T^{4} )^{4} \) |
| 47 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \) |
| 53 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \) |
| 59 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 61 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 67 | \( ( 1 - T )^{8}( 1 + T^{2} )^{4} \) |
| 71 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \) |
| 73 | \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \) |
| 79 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 83 | \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \) |
| 89 | \( ( 1 + T^{2} )^{8} \) |
| 97 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.95825131402453036523765982036, −4.70570813453337572570895833179, −4.60615088214034175762575072142, −4.48087983787020986270055012372, −4.45377581246930556150005202452, −4.27567571512133134146275240000, −3.95671247726461791859387824377, −3.94135349979347231119537022174, −3.89255125114819955002376219707, −3.87965630933932625036981791227, −3.64250115166442234441480454658, −3.35058899478422628591489559666, −3.30617896332230015490208542244, −3.05004765402552925916788092080, −3.04003834448585699471247501327, −2.51612685095028633135248256053, −2.43553094935647101183460505084, −2.24008365969667372946248075787, −2.04423283722651196890436394849, −1.97738804122392864361554238037, −1.93346830409320891915823061122, −1.74476970960032848409458642893, −1.14395072034265558483809096537, −1.08259507175776341662098159118, −0.63173197566369554946515770389,
0.63173197566369554946515770389, 1.08259507175776341662098159118, 1.14395072034265558483809096537, 1.74476970960032848409458642893, 1.93346830409320891915823061122, 1.97738804122392864361554238037, 2.04423283722651196890436394849, 2.24008365969667372946248075787, 2.43553094935647101183460505084, 2.51612685095028633135248256053, 3.04003834448585699471247501327, 3.05004765402552925916788092080, 3.30617896332230015490208542244, 3.35058899478422628591489559666, 3.64250115166442234441480454658, 3.87965630933932625036981791227, 3.89255125114819955002376219707, 3.94135349979347231119537022174, 3.95671247726461791859387824377, 4.27567571512133134146275240000, 4.45377581246930556150005202452, 4.48087983787020986270055012372, 4.60615088214034175762575072142, 4.70570813453337572570895833179, 4.95825131402453036523765982036
Plot not available for L-functions of degree greater than 10.