| L(s) = 1 | + 2·3-s + 4-s + 4·9-s − 6·11-s + 2·12-s − 4·13-s + 4·16-s − 12·17-s + 4·19-s − 6·23-s − 14·25-s + 4·27-s + 6·29-s + 4·31-s − 12·33-s + 4·36-s + 14·37-s − 8·39-s − 10·43-s − 6·44-s − 24·47-s + 8·48-s − 24·51-s − 4·52-s − 12·53-s + 8·57-s + 18·59-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/2·4-s + 4/3·9-s − 1.80·11-s + 0.577·12-s − 1.10·13-s + 16-s − 2.91·17-s + 0.917·19-s − 1.25·23-s − 2.79·25-s + 0.769·27-s + 1.11·29-s + 0.718·31-s − 2.08·33-s + 2/3·36-s + 2.30·37-s − 1.28·39-s − 1.52·43-s − 0.904·44-s − 3.50·47-s + 1.15·48-s − 3.36·51-s − 0.554·52-s − 1.64·53-s + 1.05·57-s + 2.34·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5753926190\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5753926190\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| good | 2 | $C_2^3$ | \( 1 - T^{2} - 3 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) |
| 3 | $D_4\times C_2$ | \( 1 - 2 T + 4 T^{3} - 5 T^{4} + 4 p T^{5} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2^2$ | \( ( 1 + 7 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 + 6 T + 8 T^{2} + 36 T^{3} + 267 T^{4} + 36 p T^{5} + 8 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 12 T + 77 T^{2} + 396 T^{3} + 1752 T^{4} + 396 p T^{5} + 77 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 6 T - 16 T^{2} + 36 T^{3} + 1347 T^{4} + 36 p T^{5} - 16 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 2 T + 36 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^3$ | \( 1 - 55 T^{2} + 1344 T^{4} - 55 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 10 T + 16 T^{2} - 20 T^{3} + 907 T^{4} - 20 p T^{5} + 16 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 12 T + 82 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 + 6 T + 67 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 18 T + 128 T^{2} - 1404 T^{3} + 15819 T^{4} - 1404 p T^{5} + 128 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 20 T + 205 T^{2} + 1460 T^{3} + 9904 T^{4} + 1460 p T^{5} + 205 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 2 T - 104 T^{2} + 52 T^{3} + 6907 T^{4} + 52 p T^{5} - 104 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 4 T + 123 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 22 T + 252 T^{2} - 22 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 + 6 T + 148 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 12 T - 22 T^{2} + 144 T^{3} + 6819 T^{4} + 144 p T^{5} - 22 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 8 T - 38 T^{2} - 736 T^{3} - 5213 T^{4} - 736 p T^{5} - 38 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.79421242233673463685196658529, −7.71409558411095513824980288743, −7.09664022291890803620081406570, −6.99666098866669377323165564688, −6.67770957296042179370704552244, −6.44457030137144417226037379537, −6.30269912430157453274826690170, −6.10913137510746170724150103161, −5.60359266109505575031061080483, −5.55939179360120265094307127740, −4.95818349335107414863858261755, −4.92725400131212837580603747014, −4.72153433226428100730272056951, −4.38086443015264410239308259192, −4.05646464045348324059148493748, −3.98601988788678270994597603324, −3.28333097583186957754341625540, −3.21960300161415955739881118255, −3.01807458312312908487711765801, −2.47728391969066838927809405777, −2.17384669534214914439529930006, −2.14848828465535879764944161996, −1.83054655223027217976497299863, −1.23536359908713121127715746559, −0.18071810275012759130383345859,
0.18071810275012759130383345859, 1.23536359908713121127715746559, 1.83054655223027217976497299863, 2.14848828465535879764944161996, 2.17384669534214914439529930006, 2.47728391969066838927809405777, 3.01807458312312908487711765801, 3.21960300161415955739881118255, 3.28333097583186957754341625540, 3.98601988788678270994597603324, 4.05646464045348324059148493748, 4.38086443015264410239308259192, 4.72153433226428100730272056951, 4.92725400131212837580603747014, 4.95818349335107414863858261755, 5.55939179360120265094307127740, 5.60359266109505575031061080483, 6.10913137510746170724150103161, 6.30269912430157453274826690170, 6.44457030137144417226037379537, 6.67770957296042179370704552244, 6.99666098866669377323165564688, 7.09664022291890803620081406570, 7.71409558411095513824980288743, 7.79421242233673463685196658529
