| L(s) = 1 | + 2·9-s − 16·31-s − 20·41-s − 4·49-s − 32·71-s + 16·79-s − 15·81-s + 4·89-s + 30·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 2/3·9-s − 2.87·31-s − 3.12·41-s − 4/7·49-s − 3.79·71-s + 1.80·79-s − 5/3·81-s + 0.423·89-s + 2.72·121-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 − 4·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1602577409\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1602577409\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 15 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 31 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 90 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 71 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 97 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 103 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 190 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−7.13490244593036409028054972767, −7.12095304485146774817561482633, −7.04950025673233598671835367704, −6.93688318646256783902044188879, −6.47203948165146753707819227900, −6.03387139606450548320650579275, −6.02222377050629901675630919235, −5.77409573637976002217170674408, −5.64350931980730144263433882425, −5.17826544406855739555709614605, −4.88392853711986871382091878854, −4.75987465676074394720138262528, −4.67790382624218834446385949713, −4.29174621401831268753427401147, −3.75448655624347408248338337101, −3.63322187933743533647015588480, −3.52052337192759203066512517650, −3.32238801798423529579051191639, −2.68765271082009186242206091992, −2.59912710267980451314943279094, −2.02916187817955939970491437531, −1.77169615698704252924458673297, −1.47632366415556248565461328415, −1.16575206978228968523523867777, −0.099017330053158604261749688298,
0.099017330053158604261749688298, 1.16575206978228968523523867777, 1.47632366415556248565461328415, 1.77169615698704252924458673297, 2.02916187817955939970491437531, 2.59912710267980451314943279094, 2.68765271082009186242206091992, 3.32238801798423529579051191639, 3.52052337192759203066512517650, 3.63322187933743533647015588480, 3.75448655624347408248338337101, 4.29174621401831268753427401147, 4.67790382624218834446385949713, 4.75987465676074394720138262528, 4.88392853711986871382091878854, 5.17826544406855739555709614605, 5.64350931980730144263433882425, 5.77409573637976002217170674408, 6.02222377050629901675630919235, 6.03387139606450548320650579275, 6.47203948165146753707819227900, 6.93688318646256783902044188879, 7.04950025673233598671835367704, 7.12095304485146774817561482633, 7.13490244593036409028054972767
