| L(s) = 1 | + 4·2-s + 8·4-s + 8·8-s − 4·16-s + 8·17-s − 16·23-s + 8·31-s − 32·32-s + 32·34-s − 64·46-s − 32·47-s + 32·62-s − 64·64-s + 64·68-s − 9·81-s − 128·92-s − 128·94-s − 56·113-s + 4·121-s + 64·124-s + 127-s − 64·128-s + 131-s + 64·136-s + 137-s + 139-s + 149-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 4·4-s + 2.82·8-s − 16-s + 1.94·17-s − 3.33·23-s + 1.43·31-s − 5.65·32-s + 5.48·34-s − 9.43·46-s − 4.66·47-s + 4.06·62-s − 8·64-s + 7.76·68-s − 81-s − 13.3·92-s − 13.2·94-s − 5.26·113-s + 4/11·121-s + 5.74·124-s + 0.0887·127-s − 5.65·128-s + 0.0873·131-s + 5.48·136-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.998943294\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.998943294\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^3$ | \( 1 - 142 T^{4} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 37 | $C_2^3$ | \( 1 - 1582 T^{4} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 43 | $C_2^3$ | \( 1 + 1778 T^{4} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2^3$ | \( 1 - 8302 T^{4} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.57322249823060094430394405739, −7.36020051925428138174681538814, −7.05651322915844779699268840616, −6.76035274058704159724436536967, −6.39592787552161944706199731997, −6.35914225101324827018779568778, −6.28479507670404370217445389129, −5.88118117753724617388340962091, −5.64302705373230873256554291506, −5.43268133075619471811901371932, −5.31281365473183034040497040690, −4.81872541433951315450903892771, −4.80804297653008484117947108129, −4.50580139619482215223955974534, −4.00050273246651966323171548704, −3.96558930079620413784979394887, −3.85118716125641747987445215382, −3.34081267209930714267366029809, −3.10330461774795731302091510107, −2.91580409637808775843585344815, −2.65746333312597834201702453972, −2.10387914072315863358180991277, −1.64345585494942339616386399038, −1.59752555641412705031136005861, −0.39798136038158938594505397618,
0.39798136038158938594505397618, 1.59752555641412705031136005861, 1.64345585494942339616386399038, 2.10387914072315863358180991277, 2.65746333312597834201702453972, 2.91580409637808775843585344815, 3.10330461774795731302091510107, 3.34081267209930714267366029809, 3.85118716125641747987445215382, 3.96558930079620413784979394887, 4.00050273246651966323171548704, 4.50580139619482215223955974534, 4.80804297653008484117947108129, 4.81872541433951315450903892771, 5.31281365473183034040497040690, 5.43268133075619471811901371932, 5.64302705373230873256554291506, 5.88118117753724617388340962091, 6.28479507670404370217445389129, 6.35914225101324827018779568778, 6.39592787552161944706199731997, 6.76035274058704159724436536967, 7.05651322915844779699268840616, 7.36020051925428138174681538814, 7.57322249823060094430394405739
