| L(s) = 1 | − 2·13-s + 12·17-s + 10·25-s − 12·29-s + 8·43-s + 2·49-s − 12·53-s − 4·61-s + 16·79-s − 12·101-s − 16·103-s + 24·107-s + 12·113-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 9·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 0.554·13-s + 2.91·17-s + 2·25-s − 2.22·29-s + 1.21·43-s + 2/7·49-s − 1.64·53-s − 0.512·61-s + 1.80·79-s − 1.19·101-s − 1.57·103-s + 2.32·107-s + 1.12·113-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 − 0.692·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3504384 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3504384 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.533085162\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.533085162\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 154 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−9.268123451807267301344687037471, −9.209366869335164030777677561765, −8.754121734649953833262045227067, −7.944543224238885824206823459200, −7.918041400392443925458204203521, −7.62082698432222449838988599889, −6.97813910636055363684576624627, −6.90737143399282830056499369999, −6.08989652046962761771009124632, −5.77437843594195012621508984265, −5.45263091749220973143811563412, −5.01262964210276623961677455470, −4.61039473510964536743426527653, −3.99395706974002866084525692113, −3.31003592284693946367836947711, −3.30140874401848639117646589066, −2.62737972260535610336860269209, −1.90102068319411022057398772856, −1.28076768259758631432022814307, −0.64240583080349214362919923775,
0.64240583080349214362919923775, 1.28076768259758631432022814307, 1.90102068319411022057398772856, 2.62737972260535610336860269209, 3.30140874401848639117646589066, 3.31003592284693946367836947711, 3.99395706974002866084525692113, 4.61039473510964536743426527653, 5.01262964210276623961677455470, 5.45263091749220973143811563412, 5.77437843594195012621508984265, 6.08989652046962761771009124632, 6.90737143399282830056499369999, 6.97813910636055363684576624627, 7.62082698432222449838988599889, 7.918041400392443925458204203521, 7.944543224238885824206823459200, 8.754121734649953833262045227067, 9.209366869335164030777677561765, 9.268123451807267301344687037471
