| L(s) = 1 | + 2·9-s + 12·17-s − 20·41-s − 28·49-s − 36·73-s − 15·81-s − 60·89-s + 8·97-s + 4·113-s + 34·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 24·153-s + 157-s + 163-s + 167-s + 20·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | + 2/3·9-s + 2.91·17-s − 3.12·41-s − 4·49-s − 4.21·73-s − 5/3·81-s − 6.35·89-s + 0.812·97-s + 0.376·113-s + 3.09·121-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 + 1.94·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.53·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3999080412\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3999080412\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 - 17 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 33 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 89 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 121 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
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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
−6.00833891248269748870059787458, −5.90869067126441523385798884196, −5.72834279761697451763529212966, −5.57926499046805634258681452068, −5.44751040203606988422813418196, −5.17472806567813325068534705768, −4.78286061471826263678987152399, −4.62288810544811788802928818007, −4.52718034084306128463736457329, −4.52271452759291246942313345114, −4.13631515732100051630945210583, −3.68355747511223221537375011591, −3.51115545853481729932681872616, −3.38881241923579262773212645473, −3.28912358528525055984153250031, −2.98933998591485393410476198986, −2.72442486495080174502595560066, −2.68425963713436699873147254644, −2.05374495876676979965081198916, −1.68637649933675878216714777828, −1.54639760062492613760155207057, −1.40466420492133063284002524212, −1.31011268342524137903603038099, −0.66403475056695630136571874300, −0.096441633202889356990664552413,
0.096441633202889356990664552413, 0.66403475056695630136571874300, 1.31011268342524137903603038099, 1.40466420492133063284002524212, 1.54639760062492613760155207057, 1.68637649933675878216714777828, 2.05374495876676979965081198916, 2.68425963713436699873147254644, 2.72442486495080174502595560066, 2.98933998591485393410476198986, 3.28912358528525055984153250031, 3.38881241923579262773212645473, 3.51115545853481729932681872616, 3.68355747511223221537375011591, 4.13631515732100051630945210583, 4.52271452759291246942313345114, 4.52718034084306128463736457329, 4.62288810544811788802928818007, 4.78286061471826263678987152399, 5.17472806567813325068534705768, 5.44751040203606988422813418196, 5.57926499046805634258681452068, 5.72834279761697451763529212966, 5.90869067126441523385798884196, 6.00833891248269748870059787458
