| L(s) = 1 | + 24·23-s − 14·25-s + 48·47-s + 26·49-s + 24·71-s − 44·73-s − 4·97-s + 38·121-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 | + 5.00·23-s − 2.79·25-s + 7.00·47-s + 26/7·49-s + 2.84·71-s − 5.14·73-s − 0.406·97-s + 3.45·121-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.796371883\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.796371883\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( ( 1 + 7 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 19 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 61 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 - 41 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 91 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
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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
−6.91036316619056822104799280806, −6.35187922656733768249174115453, −6.09850660379369662246111281561, −5.93025336891535594710632416453, −5.75822467184270410098237231777, −5.44937408513816121537383528075, −5.42461252454220987641683058823, −5.39244341989454252762067916813, −4.96905232129150874972531294989, −4.43229189893551821020453002881, −4.35510222263162442855299842274, −4.32467524537356462180080593115, −4.15324710468079906835343727277, −3.72653824762307117771024378138, −3.45414853250588764494975395181, −3.18984913594833498501143295009, −3.06814909938073913259813993058, −2.60667141331544139839545589736, −2.37074935412813088570220536798, −2.22591100496404850447771676019, −2.12381189676435211038764933341, −1.16107321544310654445521776653, −1.15501449662592678820165704157, −0.945192207548090539540732975535, −0.45519853778001748082868486367,
0.45519853778001748082868486367, 0.945192207548090539540732975535, 1.15501449662592678820165704157, 1.16107321544310654445521776653, 2.12381189676435211038764933341, 2.22591100496404850447771676019, 2.37074935412813088570220536798, 2.60667141331544139839545589736, 3.06814909938073913259813993058, 3.18984913594833498501143295009, 3.45414853250588764494975395181, 3.72653824762307117771024378138, 4.15324710468079906835343727277, 4.32467524537356462180080593115, 4.35510222263162442855299842274, 4.43229189893551821020453002881, 4.96905232129150874972531294989, 5.39244341989454252762067916813, 5.42461252454220987641683058823, 5.44937408513816121537383528075, 5.75822467184270410098237231777, 5.93025336891535594710632416453, 6.09850660379369662246111281561, 6.35187922656733768249174115453, 6.91036316619056822104799280806
