| L(s) = 1 | + 2·3-s − 6·7-s + 9-s + 6·11-s − 8·17-s − 6·19-s − 12·21-s + 2·23-s + 6·25-s − 2·27-s + 2·29-s + 12·33-s − 12·37-s + 36·41-s + 2·43-s + 8·49-s − 16·51-s − 12·53-s − 12·57-s + 8·61-s − 6·63-s + 42·67-s + 4·69-s − 6·71-s + 12·75-s − 36·77-s + 24·79-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 2.26·7-s + 1/3·9-s + 1.80·11-s − 1.94·17-s − 1.37·19-s − 2.61·21-s + 0.417·23-s + 6/5·25-s − 0.384·27-s + 0.371·29-s + 2.08·33-s − 1.97·37-s + 5.62·41-s + 0.304·43-s + 8/7·49-s − 2.24·51-s − 1.64·53-s − 1.58·57-s + 1.02·61-s − 0.755·63-s + 5.13·67-s + 0.481·69-s − 0.712·71-s + 1.38·75-s − 4.10·77-s + 2.70·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.843087798\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.843087798\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| good | 5 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} ) \) |
| 7 | $D_4\times C_2$ | \( 1 + 6 T + 4 p T^{2} + 96 T^{3} + 291 T^{4} + 96 p T^{5} + 4 p^{3} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 6 T + 28 T^{2} - 96 T^{3} + 267 T^{4} - 96 p T^{5} + 28 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2$$\times$$C_2^2$ | \( ( 1 + 8 T + p T^{2} )^{2}( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} ) \) |
| 19 | $D_4\times C_2$ | \( 1 + 6 T + 44 T^{2} + 192 T^{3} + 891 T^{4} + 192 p T^{5} + 44 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 2 T - 16 T^{2} + 52 T^{3} - 221 T^{4} + 52 p T^{5} - 16 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 2 T - 43 T^{2} + 22 T^{3} + 1252 T^{4} + 22 p T^{5} - 43 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 92 T^{2} + 3846 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + 48 p T^{4} + 12 p^{2} T^{5} + 85 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 36 T + 621 T^{2} - 6804 T^{3} + 51752 T^{4} - 6804 p T^{5} + 621 p^{2} T^{6} - 36 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 2 T - 8 T^{2} + 148 T^{3} - 1877 T^{4} + 148 p T^{5} - 8 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 116 T^{2} + 6810 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 6 T + 103 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 + 54 T^{2} - 565 T^{4} + 54 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 8 T - 47 T^{2} + 88 T^{3} + 4696 T^{4} + 88 p T^{5} - 47 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 42 T + 868 T^{2} - 11760 T^{3} + 113307 T^{4} - 11760 p T^{5} + 868 p^{2} T^{6} - 42 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 6 T + 148 T^{2} + 816 T^{3} + 14307 T^{4} + 816 p T^{5} + 148 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 158 T^{2} + 16131 T^{4} - 158 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 12 T + 182 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 228 T^{2} + 24074 T^{4} - 228 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 12 T + 202 T^{2} + 1848 T^{3} + 20067 T^{4} + 1848 p T^{5} + 202 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^3$ | \( 1 + 158 T^{2} + 15555 T^{4} + 158 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−7.82420297013358269313464485386, −7.24021351661811350938165437971, −7.01692139927745341119979734936, −6.89140201592224606500353821929, −6.65817814973386917527968695676, −6.55623282563754418818297559008, −6.32404254896505002427804470460, −6.20525699269522455494798904006, −5.89686578922987762295788452171, −5.50608328111035772300690237177, −5.15462137014560482086804505998, −4.93175656160080098424383277802, −4.58437145275274747665279125953, −4.18557868460119040648143612469, −3.90299633300606398718795154443, −3.84973686827284341600959328169, −3.76807867685129930061042082637, −3.27748738744588351168379374794, −2.83950675583261077305839377604, −2.57995845748586323648566589504, −2.42272070900875439720971492115, −2.27895900749039203450240028767, −1.49182821399345518166288024057, −1.09935040662624855569028054693, −0.38706085659547321548333630484,
0.38706085659547321548333630484, 1.09935040662624855569028054693, 1.49182821399345518166288024057, 2.27895900749039203450240028767, 2.42272070900875439720971492115, 2.57995845748586323648566589504, 2.83950675583261077305839377604, 3.27748738744588351168379374794, 3.76807867685129930061042082637, 3.84973686827284341600959328169, 3.90299633300606398718795154443, 4.18557868460119040648143612469, 4.58437145275274747665279125953, 4.93175656160080098424383277802, 5.15462137014560482086804505998, 5.50608328111035772300690237177, 5.89686578922987762295788452171, 6.20525699269522455494798904006, 6.32404254896505002427804470460, 6.55623282563754418818297559008, 6.65817814973386917527968695676, 6.89140201592224606500353821929, 7.01692139927745341119979734936, 7.24021351661811350938165437971, 7.82420297013358269313464485386
