| L(s) = 1 | + 2·3-s − 4-s + 3·9-s − 2·12-s + 4·13-s + 16-s − 4·17-s + 8·23-s + 4·27-s + 10·29-s − 3·36-s + 8·39-s + 8·43-s + 2·48-s + 10·49-s − 8·51-s − 4·52-s + 18·53-s + 24·61-s − 64-s + 4·68-s + 16·69-s + 10·79-s + 5·81-s + 20·87-s − 8·92-s + 4·101-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1/2·4-s + 9-s − 0.577·12-s + 1.10·13-s + 1/4·16-s − 0.970·17-s + 1.66·23-s + 0.769·27-s + 1.85·29-s − 1/2·36-s + 1.28·39-s + 1.21·43-s + 0.288·48-s + 10/7·49-s − 1.12·51-s − 0.554·52-s + 2.47·53-s + 3.07·61-s − 1/8·64-s + 0.485·68-s + 1.92·69-s + 1.12·79-s + 5/9·81-s + 2.14·87-s − 0.834·92-s + 0.398·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.499914231\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.499914231\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 57 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 117 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−8.975834818818931719184357943791, −8.973385579078192580149797827237, −8.601356744804962444876531288916, −8.578681567870534266737417827771, −7.82346884798060876949014943242, −7.57807260573556895085243077153, −6.97281249581481293356607197550, −6.77077150295250481824947855830, −6.34356398688303975406974508419, −5.78581783778706425244309741647, −5.19679800887566663338956575934, −4.96008315211531147459892897201, −4.28955877622048816617085600290, −3.95076745143106800538212320890, −3.66476397347553341140502926145, −3.01828402800010051602088117727, −2.37858492723543815766694188749, −2.33186518888172477091621796966, −1.03919931328284078250539928645, −0.952157478562862498091043618874,
0.952157478562862498091043618874, 1.03919931328284078250539928645, 2.33186518888172477091621796966, 2.37858492723543815766694188749, 3.01828402800010051602088117727, 3.66476397347553341140502926145, 3.95076745143106800538212320890, 4.28955877622048816617085600290, 4.96008315211531147459892897201, 5.19679800887566663338956575934, 5.78581783778706425244309741647, 6.34356398688303975406974508419, 6.77077150295250481824947855830, 6.97281249581481293356607197550, 7.57807260573556895085243077153, 7.82346884798060876949014943242, 8.578681567870534266737417827771, 8.601356744804962444876531288916, 8.973385579078192580149797827237, 8.975834818818931719184357943791
