L(s) = 1 | − 8·9-s − 2·25-s − 16·31-s + 16·49-s + 48·71-s + 16·79-s + 30·81-s + 24·89-s + 20·121-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 | − 8/3·9-s − 2/5·25-s − 2.87·31-s + 16/7·49-s + 5.69·71-s + 1.80·79-s + 10/3·81-s + 2.54·89-s + 1.81·121-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8069232080\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8069232080\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
good | 3 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 68 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 116 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 68 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 170 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−9.530617812293066640554114578445, −9.142138410527000982977410231135, −8.987713305828881390722723649643, −8.610490931788288429176969281261, −8.361244593007290123258692281392, −8.224443468460717456966353242797, −7.923200270208475882931251061289, −7.48179346508146734748482155634, −7.35604329280083780527848613717, −6.88400947351338394806531387559, −6.69577593568696633425737797980, −6.18496847811091275157070007980, −5.95039729332544671843986894761, −5.75733877133451644015694259716, −5.37306533955832967098114330914, −5.20126341299700026547296509883, −4.90368684188128360402585046865, −4.37624647484079732059320414888, −3.56802123144697703378578358297, −3.55917925667897857737300198247, −3.54690223371836310274677125087, −2.61331280460558680389105991696, −2.27577330372620020553451553834, −2.05406325523293706674455829453, −0.64690881897807127230974538270,
0.64690881897807127230974538270, 2.05406325523293706674455829453, 2.27577330372620020553451553834, 2.61331280460558680389105991696, 3.54690223371836310274677125087, 3.55917925667897857737300198247, 3.56802123144697703378578358297, 4.37624647484079732059320414888, 4.90368684188128360402585046865, 5.20126341299700026547296509883, 5.37306533955832967098114330914, 5.75733877133451644015694259716, 5.95039729332544671843986894761, 6.18496847811091275157070007980, 6.69577593568696633425737797980, 6.88400947351338394806531387559, 7.35604329280083780527848613717, 7.48179346508146734748482155634, 7.923200270208475882931251061289, 8.224443468460717456966353242797, 8.361244593007290123258692281392, 8.610490931788288429176969281261, 8.987713305828881390722723649643, 9.142138410527000982977410231135, 9.530617812293066640554114578445