L(s) = 1 | + 4-s + 4·9-s + 16-s + 7·19-s − 3·23-s + 2·25-s − 6·29-s + 4·36-s + 4·37-s − 15·41-s − 5·43-s + 9·47-s − 6·49-s + 3·53-s + 21·59-s + 10·61-s + 64-s + 7·76-s + 7·81-s − 24·83-s − 3·92-s − 29·97-s + 2·100-s + 15·101-s − 6·107-s + 22·109-s − 6·116-s + ⋯ |
L(s) = 1 | + 1/2·4-s + 4/3·9-s + 1/4·16-s + 1.60·19-s − 0.625·23-s + 2/5·25-s − 1.11·29-s + 2/3·36-s + 0.657·37-s − 2.34·41-s − 0.762·43-s + 1.31·47-s − 6/7·49-s + 0.412·53-s + 2.73·59-s + 1.28·61-s + 1/8·64-s + 0.802·76-s + 7/9·81-s − 2.63·83-s − 0.312·92-s − 2.94·97-s + 1/5·100-s + 1.49·101-s − 0.580·107-s + 2.10·109-s − 0.557·116-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1086652 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1086652 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.843431400\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.843431400\) |
\(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + T + p T^{2} ) \) |
| 197 | $C_2$ | \( 1 + 12 T + p T^{2} \) |
good | 3 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 73 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 49 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 130 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−8.039221873483812411388231984809, −7.51148254449544643609748174135, −7.18314935440222312352372581788, −6.76148803036768569469887860307, −6.63370411564921264674250456259, −5.65364787590285689176836751439, −5.47675226153137231695324525564, −5.08588324677145974381770960095, −4.16591985745246466832714234245, −4.10370965629658236969634818945, −3.32784914975706407969369271284, −2.92159386521356236414018807431, −2.01851681000244655595968032210, −1.62430172583941507908680617576, −0.799889792722728505245695910941,
0.799889792722728505245695910941, 1.62430172583941507908680617576, 2.01851681000244655595968032210, 2.92159386521356236414018807431, 3.32784914975706407969369271284, 4.10370965629658236969634818945, 4.16591985745246466832714234245, 5.08588324677145974381770960095, 5.47675226153137231695324525564, 5.65364787590285689176836751439, 6.63370411564921264674250456259, 6.76148803036768569469887860307, 7.18314935440222312352372581788, 7.51148254449544643609748174135, 8.039221873483812411388231984809