| L(s) = 1 | + 2·2-s + 3·4-s + 3·5-s + 4·8-s + 6·10-s − 3·11-s − 4·13-s + 5·16-s − 3·17-s − 10·19-s + 9·20-s − 6·22-s + 9·23-s + 5·25-s − 8·26-s + 6·29-s − 4·31-s + 6·32-s − 6·34-s + 4·37-s − 20·38-s + 12·40-s + 15·41-s + 43-s − 9·44-s + 18·46-s + 10·50-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.34·5-s + 1.41·8-s + 1.89·10-s − 0.904·11-s − 1.10·13-s + 5/4·16-s − 0.727·17-s − 2.29·19-s + 2.01·20-s − 1.27·22-s + 1.87·23-s + 25-s − 1.56·26-s + 1.11·29-s − 0.718·31-s + 1.06·32-s − 1.02·34-s + 0.657·37-s − 3.24·38-s + 1.89·40-s + 2.34·41-s + 0.152·43-s − 1.35·44-s + 2.65·46-s + 1.41·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63011844 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63011844 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.16667021\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.16667021\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 3 T + 16 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 3 T + 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 15 T + 130 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - T + 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 82 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 3 T + 46 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 11 T + 78 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 3 T + 70 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 7 T + 84 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 7 T + 96 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 70 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 18 T + 226 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + T + 120 T^{2} + p T^{3} + p^{2} T^{4} \) |
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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
−7.67292806725829786819046708915, −7.65398674743614503654711371186, −7.20301508429689924892448788879, −6.80814352720580831078718649464, −6.44269782175383054909055964766, −6.28751289879770305519706067953, −5.88066197145615291718335526714, −5.61707352755937392141007599464, −4.97525554524212757707466908115, −4.95837757710970986866269041544, −4.62148075616987877061448022293, −4.31794644608664716469302008939, −3.67007879400024221145789832056, −3.40728342444072256634023222954, −2.56799111915773002711713205595, −2.56278118430626572680991745555, −2.23348000867179067521673549461, −2.00888100796704150334190993698, −1.08706750673603147258075008161, −0.58079529470835522575617371732,
0.58079529470835522575617371732, 1.08706750673603147258075008161, 2.00888100796704150334190993698, 2.23348000867179067521673549461, 2.56278118430626572680991745555, 2.56799111915773002711713205595, 3.40728342444072256634023222954, 3.67007879400024221145789832056, 4.31794644608664716469302008939, 4.62148075616987877061448022293, 4.95837757710970986866269041544, 4.97525554524212757707466908115, 5.61707352755937392141007599464, 5.88066197145615291718335526714, 6.28751289879770305519706067953, 6.44269782175383054909055964766, 6.80814352720580831078718649464, 7.20301508429689924892448788879, 7.65398674743614503654711371186, 7.67292806725829786819046708915
