| L(s) = 1 | + 8·5-s + 16·17-s + 32·25-s − 32·37-s − 48·41-s − 16·53-s − 16·61-s + 24·73-s − 2·81-s + 128·85-s + 24·97-s + 48·101-s − 32·113-s + 80·121-s + 104·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 3.57·5-s + 3.88·17-s + 32/5·25-s − 5.26·37-s − 7.49·41-s − 2.19·53-s − 2.04·61-s + 2.80·73-s − 2/9·81-s + 13.8·85-s + 2.43·97-s + 4.77·101-s − 3.01·113-s + 7.27·121-s + 9.30·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.218372386\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.218372386\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + T^{4} )^{2} \) |
| 5 | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| good | 7 | \( 1 + 4 p T^{4} - 1146 T^{8} + 4 p^{5} T^{12} + p^{8} T^{16} \) |
| 11 | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 - 142 T^{4} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 8 T + 32 T^{2} - 104 T^{3} + 322 T^{4} - 104 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 36 T^{2} + 662 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( 1 - 700 T^{4} + 184518 T^{8} - 700 p^{4} T^{12} + p^{8} T^{16} \) |
| 29 | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 + 16 T + 128 T^{2} + 912 T^{3} + 6098 T^{4} + 912 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 12 T + 94 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 43 | \( 1 - 92 T^{4} - 6160986 T^{8} - 92 p^{4} T^{12} + p^{8} T^{16} \) |
| 47 | \( 1 - 5692 T^{4} + 17637894 T^{8} - 5692 p^{4} T^{12} + p^{8} T^{16} \) |
| 53 | \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( 1 + 2788 T^{4} - 12042906 T^{8} + 2788 p^{4} T^{12} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 124 T^{2} + 7782 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + 204 T^{2} + 21350 T^{4} + 204 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 12466 T^{4} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 174 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.48820259733349925057346778014, −5.47401791763435416639709600300, −5.12146012154965946890971268222, −5.03144286266106297850108938134, −5.00692857403998576349037879401, −4.95092513728393703453515116454, −4.74136163389259876954214629147, −4.60063221497993749331731080442, −4.51554978778315875007965161593, −3.77666414638299488091156137477, −3.67990078789532933993168205903, −3.57785749786600299151655869036, −3.45816334435872300551510926922, −3.28610305669540034216297133961, −3.12650384198720997292700840579, −3.12002058493849750662825070748, −3.07657551701670098253775551160, −2.38763336143636504048320041018, −2.06823761243242344176880823045, −1.98657288960737003802095389447, −1.85924119390877560639412830447, −1.70356475293786891742151314392, −1.49839896983758720461681348397, −1.37484720226801905958451630843, −0.62422920614802035164289667134,
0.62422920614802035164289667134, 1.37484720226801905958451630843, 1.49839896983758720461681348397, 1.70356475293786891742151314392, 1.85924119390877560639412830447, 1.98657288960737003802095389447, 2.06823761243242344176880823045, 2.38763336143636504048320041018, 3.07657551701670098253775551160, 3.12002058493849750662825070748, 3.12650384198720997292700840579, 3.28610305669540034216297133961, 3.45816334435872300551510926922, 3.57785749786600299151655869036, 3.67990078789532933993168205903, 3.77666414638299488091156137477, 4.51554978778315875007965161593, 4.60063221497993749331731080442, 4.74136163389259876954214629147, 4.95092513728393703453515116454, 5.00692857403998576349037879401, 5.03144286266106297850108938134, 5.12146012154965946890971268222, 5.47401791763435416639709600300, 5.48820259733349925057346778014
Plot not available for L-functions of degree greater than 10.
