| L(s) = 1 | + 8·9-s − 8·17-s − 24·49-s − 24·73-s + 30·81-s + 40·89-s − 56·97-s − 24·113-s + 28·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 64·153-s + 157-s + 163-s + 167-s + 44·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 8/3·9-s − 1.94·17-s − 3.42·49-s − 2.80·73-s + 10/3·81-s + 4.23·89-s − 5.68·97-s − 2.25·113-s + 2.54·121-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 − 5.17·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.38·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7301226798\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7301226798\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 12 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 44 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 36 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 132 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{4} \) |
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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
−6.28049786903281198309620814719, −5.98563867070134465805728374853, −5.66486331006051289567532748417, −5.52139149652929169965783667543, −5.34306830944413835717122669374, −5.07131723960889633899974597066, −4.82187019493401783324380586399, −4.54032818442999549028517649244, −4.47974711623012337104437867282, −4.27192762817440422336270687679, −4.24486946256017848547731128396, −4.06619421509131137221735559091, −3.64532367899696013383387386827, −3.39736558326103303310485540522, −3.12779900861062155119519564464, −2.95157574290098704110352043690, −2.87966636786100734241288553139, −2.17953038049223532353195875735, −2.14969421541022719032455581492, −1.85281746153931228037677546394, −1.72784655922975377498203037578, −1.39948198062476815824663434938, −1.10976750820473401848153271075, −0.74332731440726444811768072178, −0.12334894695258653574349620972,
0.12334894695258653574349620972, 0.74332731440726444811768072178, 1.10976750820473401848153271075, 1.39948198062476815824663434938, 1.72784655922975377498203037578, 1.85281746153931228037677546394, 2.14969421541022719032455581492, 2.17953038049223532353195875735, 2.87966636786100734241288553139, 2.95157574290098704110352043690, 3.12779900861062155119519564464, 3.39736558326103303310485540522, 3.64532367899696013383387386827, 4.06619421509131137221735559091, 4.24486946256017848547731128396, 4.27192762817440422336270687679, 4.47974711623012337104437867282, 4.54032818442999549028517649244, 4.82187019493401783324380586399, 5.07131723960889633899974597066, 5.34306830944413835717122669374, 5.52139149652929169965783667543, 5.66486331006051289567532748417, 5.98563867070134465805728374853, 6.28049786903281198309620814719
