| L(s) = 1 | + 2-s + 4-s + 8-s − 3·11-s + 13-s + 16-s − 3·22-s − 6·23-s − 25-s + 26-s + 32-s − 5·37-s − 3·44-s − 6·46-s + 15·47-s − 4·49-s − 50-s + 52-s − 15·59-s − 11·61-s + 64-s + 9·71-s + 13·73-s − 5·74-s + 6·83-s − 3·88-s − 6·92-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 0.904·11-s + 0.277·13-s + 1/4·16-s − 0.639·22-s − 1.25·23-s − 1/5·25-s + 0.196·26-s + 0.176·32-s − 0.821·37-s − 0.452·44-s − 0.884·46-s + 2.18·47-s − 4/7·49-s − 0.141·50-s + 0.138·52-s − 1.95·59-s − 1.40·61-s + 1/8·64-s + 1.06·71-s + 1.52·73-s − 0.581·74-s + 0.658·83-s − 0.319·88-s − 0.625·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.323386151\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.323386151\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 124 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−11.93175840452638865639316286895, −11.22182942814150452555497966112, −10.67174425204160284879494311516, −10.30857124228346755236669161181, −9.543209166229631046460181857893, −8.904370455620049484816749729107, −8.034219728011438908508765951461, −7.72054843721988559932159341952, −6.90516280942241429007092573205, −6.13383758829117159561567624241, −5.60572073573780381692137410565, −4.82484064227010511856867055970, −4.03694068669304553687817120013, −3.14740108346281397336495905178, −2.07126360645949242683072214871,
2.07126360645949242683072214871, 3.14740108346281397336495905178, 4.03694068669304553687817120013, 4.82484064227010511856867055970, 5.60572073573780381692137410565, 6.13383758829117159561567624241, 6.90516280942241429007092573205, 7.72054843721988559932159341952, 8.034219728011438908508765951461, 8.904370455620049484816749729107, 9.543209166229631046460181857893, 10.30857124228346755236669161181, 10.67174425204160284879494311516, 11.22182942814150452555497966112, 11.93175840452638865639316286895
