| L(s) = 1 | + 2·5-s − 2·11-s + 2·25-s + 16·29-s − 14·41-s + 19·49-s − 4·55-s + 4·59-s + 22·61-s − 30·71-s + 2·79-s + 18·89-s − 4·101-s − 44·109-s − 45·121-s − 6·125-s + 127-s + 131-s + 137-s + 139-s + 32·145-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.603·11-s + 2/5·25-s + 2.97·29-s − 2.18·41-s + 19/7·49-s − 0.539·55-s + 0.520·59-s + 2.81·61-s − 3.56·71-s + 0.225·79-s + 1.90·89-s − 0.398·101-s − 4.21·109-s − 4.09·121-s − 0.536·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.65·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.307·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4756021991\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4756021991\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 2 T + 2 T^{2} + 6 T^{3} - 46 T^{4} + 6 p T^{5} + 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | \( ( 1 + T^{2} )^{4} \) |
| good | 7 | \( 1 - 19 T^{2} + 158 T^{4} - 765 T^{6} + 3714 T^{8} - 765 p^{2} T^{10} + 158 p^{4} T^{12} - 19 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( ( 1 + T + 24 T^{2} + 59 T^{3} + 282 T^{4} + 59 p T^{5} + 24 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( 1 - 75 T^{2} + 2778 T^{4} - 70725 T^{6} + 1368938 T^{8} - 70725 p^{2} T^{10} + 2778 p^{4} T^{12} - 75 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( ( 1 + p T^{2} )^{8} \) |
| 23 | \( 1 - 119 T^{2} + 6898 T^{4} - 258825 T^{6} + 6944954 T^{8} - 258825 p^{2} T^{10} + 6898 p^{4} T^{12} - 119 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( ( 1 - 8 T + 96 T^{2} - 552 T^{3} + 4142 T^{4} - 552 p T^{5} + 96 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + p T^{2} )^{8} \) |
| 37 | \( 1 - 163 T^{2} + 13874 T^{4} - 795269 T^{6} + 33768826 T^{8} - 795269 p^{2} T^{10} + 13874 p^{4} T^{12} - 163 p^{6} T^{14} + p^{8} T^{16} \) |
| 41 | \( ( 1 + 7 T + 156 T^{2} + 839 T^{3} + 9426 T^{4} + 839 p T^{5} + 156 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 - 240 T^{2} + 28156 T^{4} - 2090128 T^{6} + 107021414 T^{8} - 2090128 p^{2} T^{10} + 28156 p^{4} T^{12} - 240 p^{6} T^{14} + p^{8} T^{16} \) |
| 47 | \( 1 - 312 T^{2} + 44912 T^{4} - 3894440 T^{6} + 222698718 T^{8} - 3894440 p^{2} T^{10} + 44912 p^{4} T^{12} - 312 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( 1 - 291 T^{2} + 41426 T^{4} - 70705 p T^{6} + 235496250 T^{8} - 70705 p^{3} T^{10} + 41426 p^{4} T^{12} - 291 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( ( 1 - 2 T + 150 T^{2} - 318 T^{3} + 11810 T^{4} - 318 p T^{5} + 150 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 11 T + 222 T^{2} - 1885 T^{3} + 19674 T^{4} - 1885 p T^{5} + 222 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 - 344 T^{2} + 53948 T^{4} - 5334440 T^{6} + 397158054 T^{8} - 5334440 p^{2} T^{10} + 53948 p^{4} T^{12} - 344 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( ( 1 + 15 T + 254 T^{2} + 2665 T^{3} + 27242 T^{4} + 2665 p T^{5} + 254 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( 1 - 460 T^{2} + 98468 T^{4} - 12885876 T^{6} + 1132966134 T^{8} - 12885876 p^{2} T^{10} + 98468 p^{4} T^{12} - 460 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( ( 1 - T + 140 T^{2} - 185 T^{3} + 16966 T^{4} - 185 p T^{5} + 140 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 - 200 T^{2} + 27440 T^{4} - 3205848 T^{6} + 295212222 T^{8} - 3205848 p^{2} T^{10} + 27440 p^{4} T^{12} - 200 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 9 T + 220 T^{2} - 2025 T^{3} + 23586 T^{4} - 2025 p T^{5} + 220 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 97 | \( 1 - 231 T^{2} + 32558 T^{4} - 4062681 T^{6} + 461863330 T^{8} - 4062681 p^{2} T^{10} + 32558 p^{4} T^{12} - 231 p^{6} T^{14} + p^{8} T^{16} \) |
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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
−3.33363818439993116339223807291, −3.23902353012138042802153245786, −3.18845522336633343182696634411, −2.87101805440746606160920424992, −2.85717801196002955184694949352, −2.84451677172992217417910199597, −2.74987326316847345178145792211, −2.63002211316516618451668245692, −2.52427049317184096873356979138, −2.48917265180884561962651326322, −2.14170255046019191180316135651, −2.10745206450652503191027470819, −1.98849853011643735788114876531, −1.98297838439118852706239349516, −1.81595419843381136120544762052, −1.62777787991489064041418032215, −1.43551642322668975015247826991, −1.16285698420708022022719202283, −1.11349030404002871351056000504, −1.03114653801596444840217503826, −0.972014080780196275614306854551, −0.930820802932312107105393889796, −0.53458836407882196669830257389, −0.16171433733128917064349804550, −0.079459881287751585695375181001,
0.079459881287751585695375181001, 0.16171433733128917064349804550, 0.53458836407882196669830257389, 0.930820802932312107105393889796, 0.972014080780196275614306854551, 1.03114653801596444840217503826, 1.11349030404002871351056000504, 1.16285698420708022022719202283, 1.43551642322668975015247826991, 1.62777787991489064041418032215, 1.81595419843381136120544762052, 1.98297838439118852706239349516, 1.98849853011643735788114876531, 2.10745206450652503191027470819, 2.14170255046019191180316135651, 2.48917265180884561962651326322, 2.52427049317184096873356979138, 2.63002211316516618451668245692, 2.74987326316847345178145792211, 2.84451677172992217417910199597, 2.85717801196002955184694949352, 2.87101805440746606160920424992, 3.18845522336633343182696634411, 3.23902353012138042802153245786, 3.33363818439993116339223807291
Plot not available for L-functions of degree greater than 10.
