| L(s) = 1 | + 2·2-s + 2·3-s + 2·4-s + 4·6-s + 4·7-s + 2·9-s + 2·11-s + 4·12-s + 2·13-s + 8·14-s − 4·16-s + 4·18-s − 6·19-s + 8·21-s + 4·22-s + 12·23-s + 4·26-s + 6·27-s + 8·28-s − 6·29-s − 16·31-s − 8·32-s + 4·33-s + 4·36-s − 6·37-s − 12·38-s + 4·39-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 4-s + 1.63·6-s + 1.51·7-s + 2/3·9-s + 0.603·11-s + 1.15·12-s + 0.554·13-s + 2.13·14-s − 16-s + 0.942·18-s − 1.37·19-s + 1.74·21-s + 0.852·22-s + 2.50·23-s + 0.784·26-s + 1.15·27-s + 1.51·28-s − 1.11·29-s − 2.87·31-s − 1.41·32-s + 0.696·33-s + 2/3·36-s − 0.986·37-s − 1.94·38-s + 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.922035466\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.922035466\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 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
−11.41826577559269751089365735917, −11.10877508713391411169834381874, −10.85472083963870109758320381352, −10.45165082446448936546905762564, −9.284691609793179180644047671741, −9.072584935737612026782125005921, −8.875985970573511133848777125251, −8.407825276340770772102837898559, −7.61230160143427615269243467555, −7.36049838264595022785157228476, −6.80704845056352508151051019252, −6.15207307087812482887381236306, −5.61774879205502419730075344952, −4.94240280092890009063257410052, −4.63656433274025761572374200716, −4.08217088686873098200172840342, −3.36074836278728510758095622821, −3.09042249624025882649707134560, −2.01723937967336815805471108108, −1.61522722125517201782452281131,
1.61522722125517201782452281131, 2.01723937967336815805471108108, 3.09042249624025882649707134560, 3.36074836278728510758095622821, 4.08217088686873098200172840342, 4.63656433274025761572374200716, 4.94240280092890009063257410052, 5.61774879205502419730075344952, 6.15207307087812482887381236306, 6.80704845056352508151051019252, 7.36049838264595022785157228476, 7.61230160143427615269243467555, 8.407825276340770772102837898559, 8.875985970573511133848777125251, 9.072584935737612026782125005921, 9.284691609793179180644047671741, 10.45165082446448936546905762564, 10.85472083963870109758320381352, 11.10877508713391411169834381874, 11.41826577559269751089365735917
