| L(s) = 1 | − 4·9-s + 4·11-s − 8·23-s − 2·25-s − 4·29-s − 20·37-s − 4·43-s + 4·53-s − 24·67-s − 24·71-s − 8·79-s + 7·81-s − 16·99-s + 8·107-s + 4·109-s − 24·113-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 26·169-s + ⋯ |
| L(s) = 1 | − 4/3·9-s + 1.20·11-s − 1.66·23-s − 2/5·25-s − 0.742·29-s − 3.28·37-s − 0.609·43-s + 0.549·53-s − 2.93·67-s − 2.84·71-s − 0.900·79-s + 7/9·81-s − 1.60·99-s + 0.773·107-s + 0.383·109-s − 2.25·113-s − 0.909·121-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9834496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9834496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 116 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 114 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 144 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 68 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 128 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 96 T^{2} + p^{2} T^{4} \) |
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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
−8.531916919795277141142113023168, −8.435489775209235486083589542007, −7.60578138642261970317297302566, −7.39908069533163475276064681318, −7.12624289359599945970475029205, −6.52052607861551646421173444284, −6.09662028255669746665119281822, −5.96424945078823739523838815146, −5.53021238239453348934204745277, −5.11916738278850703308263181651, −4.57931284679619351524389520305, −4.15720066599298965872326324325, −3.59033723599415064796052139010, −3.48801555724025837448906601255, −2.85137388889332375553090420188, −2.32871133200724515889016811154, −1.55810065247043291508567151287, −1.52966398953436127407182296549, 0, 0,
1.52966398953436127407182296549, 1.55810065247043291508567151287, 2.32871133200724515889016811154, 2.85137388889332375553090420188, 3.48801555724025837448906601255, 3.59033723599415064796052139010, 4.15720066599298965872326324325, 4.57931284679619351524389520305, 5.11916738278850703308263181651, 5.53021238239453348934204745277, 5.96424945078823739523838815146, 6.09662028255669746665119281822, 6.52052607861551646421173444284, 7.12624289359599945970475029205, 7.39908069533163475276064681318, 7.60578138642261970317297302566, 8.435489775209235486083589542007, 8.531916919795277141142113023168
