| L(s) = 1 | − 2·3-s + 3·9-s + 16·19-s − 8·25-s − 4·27-s + 8·31-s − 8·37-s − 8·47-s − 12·53-s − 32·57-s + 16·59-s + 16·75-s + 5·81-s + 24·83-s − 16·93-s − 8·103-s + 40·109-s + 16·111-s − 4·113-s − 14·121-s + 127-s + 131-s + 137-s + 139-s + 16·141-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 9-s + 3.67·19-s − 8/5·25-s − 0.769·27-s + 1.43·31-s − 1.31·37-s − 1.16·47-s − 1.64·53-s − 4.23·57-s + 2.08·59-s + 1.84·75-s + 5/9·81-s + 2.63·83-s − 1.65·93-s − 0.788·103-s + 3.83·109-s + 1.51·111-s − 0.376·113-s − 1.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.34·141-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22127616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22127616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.214876484\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.214876484\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 104 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 144 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 176 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
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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
−8.238355878899391202170790356117, −8.134463850449724187476184768033, −7.56911512131962067781969327562, −7.49806322855633360245194790529, −6.85373898681488259669232630995, −6.85185309899116390747198278772, −6.14299414123555015393882559570, −5.96596440518411090919316098379, −5.48192229548482792594940620930, −5.26111757836091083417399814197, −4.75509538001501434084089490232, −4.74970846225487116623024755920, −3.79863561540601503163145537388, −3.70276075977605239280744090593, −3.09077342751066770600238553071, −2.86784958058036731694398886308, −1.78589252755485389146934715695, −1.77345306485889608345278404162, −0.823550914134792127093151369627, −0.61720789054696592651770696744,
0.61720789054696592651770696744, 0.823550914134792127093151369627, 1.77345306485889608345278404162, 1.78589252755485389146934715695, 2.86784958058036731694398886308, 3.09077342751066770600238553071, 3.70276075977605239280744090593, 3.79863561540601503163145537388, 4.74970846225487116623024755920, 4.75509538001501434084089490232, 5.26111757836091083417399814197, 5.48192229548482792594940620930, 5.96596440518411090919316098379, 6.14299414123555015393882559570, 6.85185309899116390747198278772, 6.85373898681488259669232630995, 7.49806322855633360245194790529, 7.56911512131962067781969327562, 8.134463850449724187476184768033, 8.238355878899391202170790356117
