| L(s) = 1 | − 2·9-s + 16·25-s − 28·49-s + 40·73-s + 3·81-s − 24·89-s + 32·97-s + 56·113-s + 12·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 48·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 2/3·9-s + 16/5·25-s − 4·49-s + 4.68·73-s + 1/3·81-s − 2.54·89-s + 3.24·97-s + 5.26·113-s + 1.09·121-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 + 3.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.328285349\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.328285349\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 24 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 104 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
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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
−6.30156111303892850483870953887, −5.92910193474162116281494105477, −5.72312426342127197195862597049, −5.53095463387614374824168242353, −5.41286292972813491643528626143, −5.05673458612628283902143950543, −4.81994935622962626634728503357, −4.79611388457732846109673928860, −4.72211497213393291118541848319, −4.38952947145505462266646511097, −4.21603681554694101490611709564, −3.66935993015824193499090218273, −3.52494178942616757994885140449, −3.47501187046849021814280238909, −3.15154381091793796501139302291, −3.09546902043907922974688174885, −2.71911860095142834955127902418, −2.51619090944501695618862619177, −2.12923186555589464888666718699, −2.05376380654146579220679512052, −1.62755909088188818399364015141, −1.37516253127182251074843668782, −0.892128417132368433853562068890, −0.74891855722496188045815864050, −0.31952836919656524250667600605,
0.31952836919656524250667600605, 0.74891855722496188045815864050, 0.892128417132368433853562068890, 1.37516253127182251074843668782, 1.62755909088188818399364015141, 2.05376380654146579220679512052, 2.12923186555589464888666718699, 2.51619090944501695618862619177, 2.71911860095142834955127902418, 3.09546902043907922974688174885, 3.15154381091793796501139302291, 3.47501187046849021814280238909, 3.52494178942616757994885140449, 3.66935993015824193499090218273, 4.21603681554694101490611709564, 4.38952947145505462266646511097, 4.72211497213393291118541848319, 4.79611388457732846109673928860, 4.81994935622962626634728503357, 5.05673458612628283902143950543, 5.41286292972813491643528626143, 5.53095463387614374824168242353, 5.72312426342127197195862597049, 5.92910193474162116281494105477, 6.30156111303892850483870953887
