| L(s) = 1 | − 2-s + 2·4-s − 5·8-s + 3·9-s + 3·11-s + 2·13-s + 5·16-s + 7·17-s − 3·18-s − 7·19-s − 3·22-s + 6·23-s + 5·25-s − 2·26-s − 10·29-s − 10·32-s − 7·34-s + 6·36-s − 8·37-s + 7·38-s + 4·43-s + 6·44-s − 6·46-s + 7·47-s − 5·50-s + 4·52-s + 3·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 4-s − 1.76·8-s + 9-s + 0.904·11-s + 0.554·13-s + 5/4·16-s + 1.69·17-s − 0.707·18-s − 1.60·19-s − 0.639·22-s + 1.25·23-s + 25-s − 0.392·26-s − 1.85·29-s − 1.76·32-s − 1.20·34-s + 36-s − 1.31·37-s + 1.13·38-s + 0.609·43-s + 0.904·44-s − 0.884·46-s + 1.02·47-s − 0.707·50-s + 0.554·52-s + 0.412·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405769 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.651111338\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.651111338\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 7 T + 32 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 7 T + 2 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 7 T - 10 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 3 T - 58 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 6 T - 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.68458612053644865753977119388, −10.52744305472162628133655378335, −9.882074596398518959537477103817, −9.383676610850082375905670500794, −8.978064617523603637430001594600, −8.905146449742066753527586387736, −8.246256936157764178432145919971, −7.65220415425088187959310737100, −7.25806605948748998293713394908, −6.80102784244485052352179526989, −6.54522585179525654570302498294, −5.77211743066663245821371098304, −5.70699944540611899821201334029, −4.81098966747532589402369096897, −4.14720759242437579204427323377, −3.37986883239747882751581002366, −3.30887562154071612851098618355, −2.23190411749249809139188603495, −1.64588763118373417066028485251, −0.831276927847173280375774686574,
0.831276927847173280375774686574, 1.64588763118373417066028485251, 2.23190411749249809139188603495, 3.30887562154071612851098618355, 3.37986883239747882751581002366, 4.14720759242437579204427323377, 4.81098966747532589402369096897, 5.70699944540611899821201334029, 5.77211743066663245821371098304, 6.54522585179525654570302498294, 6.80102784244485052352179526989, 7.25806605948748998293713394908, 7.65220415425088187959310737100, 8.246256936157764178432145919971, 8.905146449742066753527586387736, 8.978064617523603637430001594600, 9.383676610850082375905670500794, 9.882074596398518959537477103817, 10.52744305472162628133655378335, 10.68458612053644865753977119388
