| L(s) = 1 | − 4-s + 2·7-s + 4·13-s − 3·16-s − 8·19-s + 2·25-s − 2·28-s − 8·31-s + 4·37-s − 8·43-s + 3·49-s − 4·52-s − 20·61-s + 7·64-s − 8·67-s + 28·73-s + 8·76-s + 16·79-s + 8·91-s + 28·97-s − 2·100-s − 8·103-s + 4·109-s − 6·112-s − 10·121-s + 8·124-s + 127-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 0.755·7-s + 1.10·13-s − 3/4·16-s − 1.83·19-s + 2/5·25-s − 0.377·28-s − 1.43·31-s + 0.657·37-s − 1.21·43-s + 3/7·49-s − 0.554·52-s − 2.56·61-s + 7/8·64-s − 0.977·67-s + 3.27·73-s + 0.917·76-s + 1.80·79-s + 0.838·91-s + 2.84·97-s − 1/5·100-s − 0.788·103-s + 0.383·109-s − 0.566·112-s − 0.909·121-s + 0.718·124-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7532805415\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7532805415\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 166 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−15.11316081447122050209031377631, −14.81885324888586223700140222590, −14.06926098054719753555946497209, −13.65130748536723378296433633348, −13.09139824524204740103542304769, −12.62008946584846744802697956037, −11.92091688160406750074517919675, −11.02479730124605003313336867740, −10.98695612250120843647135121140, −10.27329396446645584897172286523, −9.194175210866369558732456620981, −8.970685852658744454090100871251, −8.258563185490701385396362178951, −7.70923034018336810485556238538, −6.66210332331203199003882891704, −6.15911066831456100244343094530, −5.10078856423729307758368068237, −4.44828181879637253814536405983, −3.60365734514966092871602602517, −2.00695843206272168824484431744,
2.00695843206272168824484431744, 3.60365734514966092871602602517, 4.44828181879637253814536405983, 5.10078856423729307758368068237, 6.15911066831456100244343094530, 6.66210332331203199003882891704, 7.70923034018336810485556238538, 8.258563185490701385396362178951, 8.970685852658744454090100871251, 9.194175210866369558732456620981, 10.27329396446645584897172286523, 10.98695612250120843647135121140, 11.02479730124605003313336867740, 11.92091688160406750074517919675, 12.62008946584846744802697956037, 13.09139824524204740103542304769, 13.65130748536723378296433633348, 14.06926098054719753555946497209, 14.81885324888586223700140222590, 15.11316081447122050209031377631
