L(s) = 1 | + 8-s + 3·11-s + 3·41-s − 3·43-s − 6·59-s − 3·67-s + 3·88-s − 3·89-s − 3·97-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
L(s) = 1 | + 8-s + 3·11-s + 3·41-s − 3·43-s − 6·59-s − 3·67-s + 3·88-s − 3·89-s − 3·97-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7489681680\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7489681680\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T^{3} + T^{6} \) |
| 3 | \( 1 \) |
good | 5 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 7 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 11 | \( ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \) |
| 13 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 17 | \( ( 1 - T^{3} + T^{6} )^{2} \) |
| 19 | \( ( 1 + T^{3} + T^{6} )^{2} \) |
| 23 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 29 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 31 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 37 | \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \) |
| 41 | \( ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \) |
| 43 | \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \) |
| 47 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 53 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 59 | \( ( 1 + T )^{6}( 1 - T^{3} + T^{6} ) \) |
| 61 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 67 | \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \) |
| 71 | \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \) |
| 73 | \( ( 1 + T^{3} + T^{6} )^{2} \) |
| 79 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 83 | \( ( 1 - T^{3} + T^{6} )^{2} \) |
| 89 | \( ( 1 + T )^{6}( 1 - T + T^{2} )^{3} \) |
| 97 | \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.98450871464762345955013208741, −5.80841108431118560691542198890, −5.60850636494450442091549474917, −5.57865849778524076113325276868, −5.08936252215798086106877877919, −4.90411437313865413709001679114, −4.76728833577939548002084748431, −4.64058123968197968418501812715, −4.51387278780308344635790454175, −4.36269865603094444550033539475, −4.27009057692490596875312490315, −3.84692162109391219141052490840, −3.81137319305538519004421568370, −3.76314719529268763888071934575, −3.45490098698584044943874717969, −3.04572850693123334834849917021, −2.94297834986848735156757820177, −2.84081477783645816140093298989, −2.78836931095467073363034603674, −2.15089482639313799589237408829, −1.89428405712027679101302479928, −1.70823222980527210448508480564, −1.37093533327596151907169413100, −1.34644698177982694049245314355, −1.20756043082390257455401919427,
1.20756043082390257455401919427, 1.34644698177982694049245314355, 1.37093533327596151907169413100, 1.70823222980527210448508480564, 1.89428405712027679101302479928, 2.15089482639313799589237408829, 2.78836931095467073363034603674, 2.84081477783645816140093298989, 2.94297834986848735156757820177, 3.04572850693123334834849917021, 3.45490098698584044943874717969, 3.76314719529268763888071934575, 3.81137319305538519004421568370, 3.84692162109391219141052490840, 4.27009057692490596875312490315, 4.36269865603094444550033539475, 4.51387278780308344635790454175, 4.64058123968197968418501812715, 4.76728833577939548002084748431, 4.90411437313865413709001679114, 5.08936252215798086106877877919, 5.57865849778524076113325276868, 5.60850636494450442091549474917, 5.80841108431118560691542198890, 5.98450871464762345955013208741
Plot not available for L-functions of degree greater than 10.