| L(s) = 1 | − 3·5-s + 2·7-s − 5·9-s + 2·11-s + 4·13-s − 10·17-s + 3·19-s − 6·23-s + 6·25-s − 2·27-s + 14·29-s + 10·31-s − 6·35-s + 2·37-s − 20·41-s + 10·43-s + 15·45-s − 6·47-s − 9·49-s + 10·53-s − 6·55-s − 6·59-s + 18·61-s − 10·63-s − 12·65-s − 2·67-s + 6·71-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.755·7-s − 5/3·9-s + 0.603·11-s + 1.10·13-s − 2.42·17-s + 0.688·19-s − 1.25·23-s + 6/5·25-s − 0.384·27-s + 2.59·29-s + 1.79·31-s − 1.01·35-s + 0.328·37-s − 3.12·41-s + 1.52·43-s + 2.23·45-s − 0.875·47-s − 9/7·49-s + 1.37·53-s − 0.809·55-s − 0.781·59-s + 2.30·61-s − 1.25·63-s − 1.48·65-s − 0.244·67-s + 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.911741055\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.911741055\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) |
| 19 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + 5 T^{2} + 2 T^{3} + 5 p T^{4} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 2 T + 13 T^{2} - 32 T^{3} + 13 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 2 T + 29 T^{2} - 40 T^{3} + 29 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 4 T + 41 T^{2} - 102 T^{3} + 41 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 10 T + 63 T^{2} + 300 T^{3} + 63 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 6 T + 53 T^{2} + 176 T^{3} + 53 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 14 T + 99 T^{2} - 516 T^{3} + 99 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 10 T + 105 T^{2} - 580 T^{3} + 105 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 2 T + p T^{2} - 2 p T^{3} + p^{2} T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 20 T + 219 T^{2} + 1608 T^{3} + 219 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 10 T + 157 T^{2} - 880 T^{3} + 157 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 6 T + 125 T^{2} + 464 T^{3} + 125 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 10 T + 133 T^{2} - 726 T^{3} + 133 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 6 T + 69 T^{2} + 492 T^{3} + 69 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 18 T + 243 T^{2} - 2144 T^{3} + 243 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 2 T + 137 T^{2} + 354 T^{3} + 137 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 6 T + 65 T^{2} - 1148 T^{3} + 65 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 2 T + 71 T^{2} - 588 T^{3} + 71 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 18 T + 225 T^{2} - 2324 T^{3} + 225 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 2 T + 157 T^{2} + 600 T^{3} + 157 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 2 T + 183 T^{2} + 588 T^{3} + 183 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 6 T + 93 T^{2} - 1218 T^{3} + 93 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.94499160487361270392177516617, −6.79731248425762618291820850515, −6.71796391755918363124243533529, −6.43806867110891013117489003454, −6.32789508637320970201325356103, −5.95118314181983831190784966385, −5.82926439485615935538994265888, −5.33719870713660144768160273847, −5.09831990451377708266712486174, −4.96871982537810166180751571418, −4.58971582281570736643903657065, −4.38082524438121947896708005647, −4.35587608230636183323694379809, −3.80291470520176358431758100948, −3.71260367545991939151686305059, −3.41105919492567194521931606125, −3.02504845998818992758739209627, −2.86801915794844388680296117320, −2.59357150607033699117566629479, −2.08684099872866266532992437071, −1.91513244637481274119055830575, −1.60296479857035759748867218883, −0.899616230530764158405761609521, −0.63106673901661671720498497482, −0.44016375505939546924788901505,
0.44016375505939546924788901505, 0.63106673901661671720498497482, 0.899616230530764158405761609521, 1.60296479857035759748867218883, 1.91513244637481274119055830575, 2.08684099872866266532992437071, 2.59357150607033699117566629479, 2.86801915794844388680296117320, 3.02504845998818992758739209627, 3.41105919492567194521931606125, 3.71260367545991939151686305059, 3.80291470520176358431758100948, 4.35587608230636183323694379809, 4.38082524438121947896708005647, 4.58971582281570736643903657065, 4.96871982537810166180751571418, 5.09831990451377708266712486174, 5.33719870713660144768160273847, 5.82926439485615935538994265888, 5.95118314181983831190784966385, 6.32789508637320970201325356103, 6.43806867110891013117489003454, 6.71796391755918363124243533529, 6.79731248425762618291820850515, 6.94499160487361270392177516617
