| L(s) = 1 | + 3·2-s + 6·4-s + 10·8-s + 15·16-s − 3·17-s + 3·25-s − 27-s + 21·32-s − 9·34-s − 3·43-s + 3·49-s + 9·50-s − 3·54-s + 28·64-s − 18·68-s − 9·86-s − 3·89-s + 9·98-s + 18·100-s − 3·107-s − 6·108-s + 127-s + 36·128-s + 131-s − 30·136-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 3·2-s + 6·4-s + 10·8-s + 15·16-s − 3·17-s + 3·25-s − 27-s + 21·32-s − 9·34-s − 3·43-s + 3·49-s + 9·50-s − 3·54-s + 28·64-s − 18·68-s − 9·86-s − 3·89-s + 9·98-s + 18·100-s − 3·107-s − 6·108-s + 127-s + 36·128-s + 131-s − 30·136-s + 137-s + 139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(13.10234539\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.10234539\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 19 | | \( 1 \) |
| good | 3 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 11 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 17 | $C_2$ | \( ( 1 + T + T^{2} )^{3} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 41 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 43 | $C_2$ | \( ( 1 + T + T^{2} )^{3} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 59 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 67 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 73 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 83 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 89 | $C_2$ | \( ( 1 + T + T^{2} )^{3} \) |
| 97 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
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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
−7.81637811177710397698344698690, −7.48820981806210813748944614009, −7.06049887508555649186203211521, −6.88837614757742351153368246457, −6.82726721466616572431892745613, −6.69867587051451413250620157227, −6.57841628931585159393089844289, −5.86952057508870356434587931427, −5.81115619292907931330242607818, −5.74239784395384400241760929041, −5.10571532356080610418815195247, −5.02136434763439624220535769718, −4.88139765214673274116053061492, −4.50115405291428778044115903668, −4.18980652204260096263471002782, −4.15256652945481146329480877384, −3.74839365791484164493186230138, −3.43115106149665841716232510825, −3.07924806086391382239098424675, −2.70781819772934452087379043192, −2.56620910063629063395810546989, −2.29866719919812353434699277083, −1.84410440710171532302524274272, −1.52021723129955634408085282500, −1.06885549366255180848247068015,
1.06885549366255180848247068015, 1.52021723129955634408085282500, 1.84410440710171532302524274272, 2.29866719919812353434699277083, 2.56620910063629063395810546989, 2.70781819772934452087379043192, 3.07924806086391382239098424675, 3.43115106149665841716232510825, 3.74839365791484164493186230138, 4.15256652945481146329480877384, 4.18980652204260096263471002782, 4.50115405291428778044115903668, 4.88139765214673274116053061492, 5.02136434763439624220535769718, 5.10571532356080610418815195247, 5.74239784395384400241760929041, 5.81115619292907931330242607818, 5.86952057508870356434587931427, 6.57841628931585159393089844289, 6.69867587051451413250620157227, 6.82726721466616572431892745613, 6.88837614757742351153368246457, 7.06049887508555649186203211521, 7.48820981806210813748944614009, 7.81637811177710397698344698690
