| L(s) = 1 | + 16·19-s − 8·25-s − 32·43-s + 4·49-s + 16·67-s − 16·73-s + 32·97-s + 28·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 28·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | + 3.67·19-s − 8/5·25-s − 4.87·43-s + 4/7·49-s + 1.95·67-s − 1.87·73-s + 3.24·97-s + 2.54·121-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 + 2.15·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.825082942\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.825082942\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 52 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 80 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 52 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 146 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 128 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
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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
−8.397330527421973118006201571349, −8.243470306468165101096091355078, −8.227849519968208422563471102326, −7.74768226398637889560513790161, −7.69416119758337514747717904456, −7.31423318661019843731643420217, −6.93124163591534699241382558603, −6.87351761499343924902840315998, −6.74458829304317199821371905847, −6.12341103912649093326759272255, −5.82690038813548006763938626442, −5.70645063116823257688276272452, −5.42307176504799492297468197603, −5.04799983282281827033846767857, −4.86337652933367729055055319932, −4.63907974551564495816861100585, −4.08343408926785093362318250365, −3.74312274981843969306408564275, −3.25484984870940174907580217689, −3.16592346264426501191292936805, −3.14539481781211993481595235758, −2.17511110363757104795173775784, −1.88993731868478107663922835759, −1.42902529885046489985179249064, −0.68380938467871172893496655527,
0.68380938467871172893496655527, 1.42902529885046489985179249064, 1.88993731868478107663922835759, 2.17511110363757104795173775784, 3.14539481781211993481595235758, 3.16592346264426501191292936805, 3.25484984870940174907580217689, 3.74312274981843969306408564275, 4.08343408926785093362318250365, 4.63907974551564495816861100585, 4.86337652933367729055055319932, 5.04799983282281827033846767857, 5.42307176504799492297468197603, 5.70645063116823257688276272452, 5.82690038813548006763938626442, 6.12341103912649093326759272255, 6.74458829304317199821371905847, 6.87351761499343924902840315998, 6.93124163591534699241382558603, 7.31423318661019843731643420217, 7.69416119758337514747717904456, 7.74768226398637889560513790161, 8.227849519968208422563471102326, 8.243470306468165101096091355078, 8.397330527421973118006201571349
