| L(s) = 1 | + 2·3-s + 3·9-s − 4·11-s − 4·13-s − 8·17-s − 4·23-s + 2·25-s + 4·27-s + 4·29-s − 8·31-s − 8·33-s − 4·37-s − 8·39-s − 16·41-s + 16·43-s + 8·47-s − 16·51-s − 4·53-s − 16·59-s + 4·61-s − 8·67-s − 8·69-s − 12·71-s − 12·73-s + 4·75-s + 8·79-s + 5·81-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 9-s − 1.20·11-s − 1.10·13-s − 1.94·17-s − 0.834·23-s + 2/5·25-s + 0.769·27-s + 0.742·29-s − 1.43·31-s − 1.39·33-s − 0.657·37-s − 1.28·39-s − 2.49·41-s + 2.43·43-s + 1.16·47-s − 2.24·51-s − 0.549·53-s − 2.08·59-s + 0.512·61-s − 0.977·67-s − 0.963·69-s − 1.42·71-s − 1.40·73-s + 0.461·75-s + 0.900·79-s + 5/9·81-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 16 T + 134 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 62 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 16 T + 134 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 166 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−8.126069212150781829347633505375, −7.65655153519014438300506913148, −7.47939256714139402939178667634, −7.19177313172231055854087072298, −6.87203165253328850354091897905, −6.32672890790781071327058228473, −5.95651784208346914267944812173, −5.58924120207586914622275561196, −4.88546050071554646017820013270, −4.76500196871709101852298642118, −4.50871196571380083253695256139, −3.83622726840416230445649723170, −3.60166912908466202441493709313, −2.92315742196825821147110147055, −2.59050975081545585920328655694, −2.35320073375944774010318320275, −1.84387063364667544303514038031, −1.32913423979140366753743989114, 0, 0,
1.32913423979140366753743989114, 1.84387063364667544303514038031, 2.35320073375944774010318320275, 2.59050975081545585920328655694, 2.92315742196825821147110147055, 3.60166912908466202441493709313, 3.83622726840416230445649723170, 4.50871196571380083253695256139, 4.76500196871709101852298642118, 4.88546050071554646017820013270, 5.58924120207586914622275561196, 5.95651784208346914267944812173, 6.32672890790781071327058228473, 6.87203165253328850354091897905, 7.19177313172231055854087072298, 7.47939256714139402939178667634, 7.65655153519014438300506913148, 8.126069212150781829347633505375
