| L(s) = 1 | − 2·4-s − 4·5-s − 2·9-s − 5·16-s + 8·20-s + 2·25-s + 4·36-s + 8·45-s + 28·49-s + 20·64-s + 48·71-s + 20·80-s + 3·81-s − 24·89-s − 4·100-s + 28·125-s + 127-s + 131-s + 137-s + 139-s + 10·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 12·169-s + ⋯ |
| L(s) = 1 | − 4-s − 1.78·5-s − 2/3·9-s − 5/4·16-s + 1.78·20-s + 2/5·25-s + 2/3·36-s + 1.19·45-s + 4·49-s + 5/2·64-s + 5.69·71-s + 2.23·80-s + 1/3·81-s − 2.54·89-s − 2/5·100-s + 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5/6·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.923·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2468431486\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2468431486\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( ( 1 + T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2} \) |
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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
−6.75224143866126394330193215536, −6.26552584392574081616624094013, −6.22975913797124368422794781131, −5.88693765268070402564711444577, −5.80908696434550465916362382156, −5.39488158965215705124620683264, −5.23819242570815352755350787687, −5.13313972688726059222824635259, −4.74297570315571309170424273559, −4.60925712704995932404702633081, −4.47653462927509338405714176869, −4.00044491987053507799136919269, −3.92203046049116926398319652469, −3.84700129119144585326174465449, −3.63651926977594442312264451213, −3.41478905759745497837594884478, −2.99690338586415060694888931475, −2.62042364449397793169985674253, −2.37601141727759994178201477679, −2.16911101045342507307724969054, −2.13216109295891719054374850049, −1.27889281267816482585358764778, −0.982675863704290588817143836424, −0.60361635628708093283541585451, −0.15212646579063749017579757646,
0.15212646579063749017579757646, 0.60361635628708093283541585451, 0.982675863704290588817143836424, 1.27889281267816482585358764778, 2.13216109295891719054374850049, 2.16911101045342507307724969054, 2.37601141727759994178201477679, 2.62042364449397793169985674253, 2.99690338586415060694888931475, 3.41478905759745497837594884478, 3.63651926977594442312264451213, 3.84700129119144585326174465449, 3.92203046049116926398319652469, 4.00044491987053507799136919269, 4.47653462927509338405714176869, 4.60925712704995932404702633081, 4.74297570315571309170424273559, 5.13313972688726059222824635259, 5.23819242570815352755350787687, 5.39488158965215705124620683264, 5.80908696434550465916362382156, 5.88693765268070402564711444577, 6.22975913797124368422794781131, 6.26552584392574081616624094013, 6.75224143866126394330193215536
