| L(s) = 1 | + 4·25-s + 20·49-s + 56·73-s + 8·97-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 + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | + 4/5·25-s + 20/7·49-s + 6.55·73-s + 0.812·97-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 + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.765247689\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.765247689\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 + 94 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 134 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 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
−6.43690853373453859961475501483, −6.21397450754112956338187884275, −5.87791259663938372275636224735, −5.79646602759485350974595448665, −5.70396727625528097832101974596, −5.23788072217281501301061274510, −5.04415407060680146545869412749, −5.02755937968520251162104889971, −4.76938310557008972052183149330, −4.60657202727711500611385483625, −4.15588544064717398883299464868, −3.99761732076258747013196019973, −3.83985514106154455783933479406, −3.56534460915115997821247876182, −3.40032134506040224179197742817, −3.16816409789108797873153323968, −2.81221082798490615191322599389, −2.47337821801203215015921830627, −2.20547062710939264192158767045, −2.16917135712474218619617438038, −1.97306422197523117783472654739, −1.20672604666220571211921319724, −0.991893106831129786270058503736, −0.938690730154911277958600849100, −0.29243071556490372072572963542,
0.29243071556490372072572963542, 0.938690730154911277958600849100, 0.991893106831129786270058503736, 1.20672604666220571211921319724, 1.97306422197523117783472654739, 2.16917135712474218619617438038, 2.20547062710939264192158767045, 2.47337821801203215015921830627, 2.81221082798490615191322599389, 3.16816409789108797873153323968, 3.40032134506040224179197742817, 3.56534460915115997821247876182, 3.83985514106154455783933479406, 3.99761732076258747013196019973, 4.15588544064717398883299464868, 4.60657202727711500611385483625, 4.76938310557008972052183149330, 5.02755937968520251162104889971, 5.04415407060680146545869412749, 5.23788072217281501301061274510, 5.70396727625528097832101974596, 5.79646602759485350974595448665, 5.87791259663938372275636224735, 6.21397450754112956338187884275, 6.43690853373453859961475501483
