| L(s) = 1 | − 8·11-s + 4·13-s − 8·23-s + 8·25-s + 16·37-s + 24·47-s + 6·49-s + 16·61-s − 8·71-s − 16·73-s + 24·83-s + 28·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s − 32·143-s + 149-s + 151-s + 157-s + 163-s + 167-s − 14·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 2.41·11-s + 1.10·13-s − 1.66·23-s + 8/5·25-s + 2.63·37-s + 3.50·47-s + 6/7·49-s + 2.04·61-s − 0.949·71-s − 1.87·73-s + 2.63·83-s + 2.68·109-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.67·143-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.07·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.141090536\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.141090536\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 64 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 56 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 64 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
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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
−9.054391760774999360260587449735, −8.854265566896114308633366959102, −8.271091592999289937047171871556, −8.155284680866821212411763501806, −7.61293359811443231651344440672, −7.41735531102962315051526068878, −7.04329114412584343484852007096, −6.28208328102252349530996046477, −5.98180406864672585525847776756, −5.80247300573540949560544946006, −5.24159772156522462403328546607, −4.92407014084292868688175349228, −4.19799100720704853230427825494, −4.13883471506571946281117468663, −3.41797719529731272497047104348, −2.81830461708124090825308990807, −2.43209672067615054521389300493, −2.18220387070786472842303972083, −1.06425319391936915613189201724, −0.58277542301415223049391014972,
0.58277542301415223049391014972, 1.06425319391936915613189201724, 2.18220387070786472842303972083, 2.43209672067615054521389300493, 2.81830461708124090825308990807, 3.41797719529731272497047104348, 4.13883471506571946281117468663, 4.19799100720704853230427825494, 4.92407014084292868688175349228, 5.24159772156522462403328546607, 5.80247300573540949560544946006, 5.98180406864672585525847776756, 6.28208328102252349530996046477, 7.04329114412584343484852007096, 7.41735531102962315051526068878, 7.61293359811443231651344440672, 8.155284680866821212411763501806, 8.271091592999289937047171871556, 8.854265566896114308633366959102, 9.054391760774999360260587449735
