| L(s) = 1 | − 9-s + 2·11-s + 16·19-s + 12·29-s − 4·41-s + 10·49-s + 24·59-s − 12·61-s − 8·79-s + 81-s + 20·89-s − 2·99-s − 4·101-s − 28·109-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s − 16·171-s + 173-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 0.603·11-s + 3.67·19-s + 2.22·29-s − 0.624·41-s + 10/7·49-s + 3.12·59-s − 1.53·61-s − 0.900·79-s + 1/9·81-s + 2.11·89-s − 0.201·99-s − 0.398·101-s − 2.68·109-s + 3/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2·169-s − 1.22·171-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.171943042\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.171943042\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−8.218521616980910281565085030975, −7.70244499336281913361375763585, −7.61839975375642601302324527177, −7.00190949550921785669093930579, −6.72237497000611550431506246506, −6.70883040759150333342930480010, −5.92862147463046249414199029981, −5.56868606029200738402527268971, −5.48297414910302520296289474791, −4.99235516590005760948603119913, −4.66882319100171949225243533082, −4.17367255792814338863269024038, −3.75097053385369957048358600555, −3.21614378908363478835842933461, −3.14119440342357911374726679775, −2.59232967348681751526873376582, −2.16431386271444548306017826196, −1.23882157580044949334876803780, −1.16042143986838353801238030376, −0.59895102003418578682316630807,
0.59895102003418578682316630807, 1.16042143986838353801238030376, 1.23882157580044949334876803780, 2.16431386271444548306017826196, 2.59232967348681751526873376582, 3.14119440342357911374726679775, 3.21614378908363478835842933461, 3.75097053385369957048358600555, 4.17367255792814338863269024038, 4.66882319100171949225243533082, 4.99235516590005760948603119913, 5.48297414910302520296289474791, 5.56868606029200738402527268971, 5.92862147463046249414199029981, 6.70883040759150333342930480010, 6.72237497000611550431506246506, 7.00190949550921785669093930579, 7.61839975375642601302324527177, 7.70244499336281913361375763585, 8.218521616980910281565085030975
