| L(s) = 1 | − 5-s + 3·7-s + 5·11-s + 5·13-s + 4·17-s − 8·19-s − 23-s + 5·25-s − 9·29-s + 31-s − 3·35-s − 12·37-s + 3·41-s − 43-s − 3·47-s + 7·49-s − 4·53-s − 5·55-s + 11·59-s − 7·61-s − 5·65-s + 67-s − 8·71-s − 4·73-s + 15·77-s − 79-s + 83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.13·7-s + 1.50·11-s + 1.38·13-s + 0.970·17-s − 1.83·19-s − 0.208·23-s + 25-s − 1.67·29-s + 0.179·31-s − 0.507·35-s − 1.97·37-s + 0.468·41-s − 0.152·43-s − 0.437·47-s + 49-s − 0.549·53-s − 0.674·55-s + 1.43·59-s − 0.896·61-s − 0.620·65-s + 0.122·67-s − 0.949·71-s − 0.468·73-s + 1.70·77-s − 0.112·79-s + 0.109·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.580576737\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.580576737\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 9 T + 52 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - T - 30 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 3 T - 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 11 T + 62 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - T - 66 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - T - 82 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 13 T + 72 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.33301225841225625353032989756, −12.08951976212825155123198909488, −11.54674992129962279971637330726, −11.19138873605400697127518679357, −10.54941577732228680779928163285, −10.52469225284218391935921330481, −9.482776687426013354812159507706, −8.999481552695027050128102820440, −8.428057998910143146019568767755, −8.421991505255167162476740288671, −7.47582765709396604650330576333, −7.12232559630798562910267941350, −6.21757808552297405855662949323, −6.08959875774201484024840710972, −5.10502125316642197607072094461, −4.55642019763593254059847043674, −3.67068864511486347733686858899, −3.62592093821060769785626162041, −2.03593968508812365837195667138, −1.30484125328406603231345639004,
1.30484125328406603231345639004, 2.03593968508812365837195667138, 3.62592093821060769785626162041, 3.67068864511486347733686858899, 4.55642019763593254059847043674, 5.10502125316642197607072094461, 6.08959875774201484024840710972, 6.21757808552297405855662949323, 7.12232559630798562910267941350, 7.47582765709396604650330576333, 8.421991505255167162476740288671, 8.428057998910143146019568767755, 8.999481552695027050128102820440, 9.482776687426013354812159507706, 10.52469225284218391935921330481, 10.54941577732228680779928163285, 11.19138873605400697127518679357, 11.54674992129962279971637330726, 12.08951976212825155123198909488, 12.33301225841225625353032989756
