| L(s) = 1 | + 10·7-s − 3·9-s + 4·16-s − 14·19-s + 14·31-s + 2·37-s − 6·43-s + 59·49-s − 30·63-s − 32·67-s + 34·73-s + 38·97-s − 38·109-s + 40·112-s + 127-s + 131-s − 140·133-s + 137-s + 139-s − 12·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 42·171-s + 173-s + ⋯ |
| L(s) = 1 | + 3.77·7-s − 9-s + 16-s − 3.21·19-s + 2.51·31-s + 0.328·37-s − 0.914·43-s + 59/7·49-s − 3.77·63-s − 3.90·67-s + 3.97·73-s + 3.85·97-s − 3.63·109-s + 3.77·112-s + 0.0887·127-s + 0.0873·131-s − 12.1·133-s + 0.0854·137-s + 0.0848·139-s − 144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.21·171-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.659169431\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.659169431\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} )( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} ) \) |
| 5 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 7 | $C_2$$\times$$C_2^2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 2 T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$$\times$$C_2^2$ | \( ( 1 + 7 T + p T^{2} )^{2}( 1 - 37 T^{2} + p^{2} T^{4} ) \) |
| 23 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$$\times$$C_2^2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 - 13 T^{2} + p^{2} T^{4} ) \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + p T^{2} )^{2}( 1 + 47 T^{2} + p^{2} T^{4} ) \) |
| 41 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 59 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 61 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 121 T^{2} + p^{2} T^{4} )( 1 + 74 T^{2} + p^{2} T^{4} ) \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + 16 T + p T^{2} )^{2}( 1 - 109 T^{2} + p^{2} T^{4} ) \) |
| 71 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 73 | $C_2$$\times$$C_2^2$ | \( ( 1 - 17 T + p T^{2} )^{2}( 1 + 143 T^{2} + p^{2} T^{4} ) \) |
| 79 | $C_2^2$ | \( ( 1 + 11 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 - 19 T + p T^{2} )^{2}( 1 + 2 T^{2} + p^{2} T^{4} ) \) |
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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
−8.083475485732295633049417322350, −7.80534587103804778134967012281, −7.43157918959302195069625635514, −7.29796146856845423946532991222, −7.12665778599877738550204658563, −6.35320255102913005744137941256, −6.32926387454556040544914195672, −6.21119680139012475574013448000, −6.15074601469211428034605277285, −5.36430651755758200369668436988, −5.32239260050040866946285818753, −5.25939419176201512386259936420, −4.63890012588682189311900963985, −4.59688469167646466516192738969, −4.59243241489671934249768490685, −4.12378034154647058043967478489, −3.85515500815609158984814522605, −3.50282289698192908640937145607, −2.96815786718822179093123457294, −2.44360027744202605357775370412, −2.42371964616249924466544536967, −2.05299892339382371345659382844, −1.48776120123247989829559681577, −1.40453528615404215089760891072, −0.64308601347123410262181001132,
0.64308601347123410262181001132, 1.40453528615404215089760891072, 1.48776120123247989829559681577, 2.05299892339382371345659382844, 2.42371964616249924466544536967, 2.44360027744202605357775370412, 2.96815786718822179093123457294, 3.50282289698192908640937145607, 3.85515500815609158984814522605, 4.12378034154647058043967478489, 4.59243241489671934249768490685, 4.59688469167646466516192738969, 4.63890012588682189311900963985, 5.25939419176201512386259936420, 5.32239260050040866946285818753, 5.36430651755758200369668436988, 6.15074601469211428034605277285, 6.21119680139012475574013448000, 6.32926387454556040544914195672, 6.35320255102913005744137941256, 7.12665778599877738550204658563, 7.29796146856845423946532991222, 7.43157918959302195069625635514, 7.80534587103804778134967012281, 8.083475485732295633049417322350
