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
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3308949430\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3308949430\) |
\(L(1)\) |
|
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
−8.253896786488419134638259316386, −7.69045723559412709775374601597, −7.48526290844213018085038823308, −7.29645500879492379088501873486, −6.98119235312974359389896365805, −6.81708427976205648696205987303, −6.57370698852761472192408807411, −6.42733388039291255231291936976, −6.30386950254554263872415327435, −5.93106500182298703401253164690, −5.32575182473505568063069451493, −5.28674696023220523436484581613, −4.91259961683046460173713252046, −4.57847905522829051354046089033, −4.16207856650570433711671946561, −3.84078533487694991834191055360, −3.11878518588072946754418955709, −3.06077957811915668233296723724, −3.04515974026068900585476587525, −2.27439455309821519325338217178, −2.12604960266300348604347603922, −2.05245544501310438535749170104, −1.42546791126607998135656767353, −0.898999948754934995127462865547, −0.59202279302345360319803236888,
0.59202279302345360319803236888, 0.898999948754934995127462865547, 1.42546791126607998135656767353, 2.05245544501310438535749170104, 2.12604960266300348604347603922, 2.27439455309821519325338217178, 3.04515974026068900585476587525, 3.06077957811915668233296723724, 3.11878518588072946754418955709, 3.84078533487694991834191055360, 4.16207856650570433711671946561, 4.57847905522829051354046089033, 4.91259961683046460173713252046, 5.28674696023220523436484581613, 5.32575182473505568063069451493, 5.93106500182298703401253164690, 6.30386950254554263872415327435, 6.42733388039291255231291936976, 6.57370698852761472192408807411, 6.81708427976205648696205987303, 6.98119235312974359389896365805, 7.29645500879492379088501873486, 7.48526290844213018085038823308, 7.69045723559412709775374601597, 8.253896786488419134638259316386