L(s) = 1 | − 2-s + 5·3-s − 5·6-s − 7-s + 15·9-s + 5·11-s + 5·13-s + 14-s − 17-s − 15·18-s − 5·21-s − 5·22-s − 2·23-s − 25-s − 5·26-s + 35·27-s − 2·31-s + 25·33-s + 34-s + 25·39-s + 5·42-s + 2·46-s + 50-s − 5·51-s − 2·53-s − 35·54-s + 2·62-s + ⋯ |
L(s) = 1 | − 2-s + 5·3-s − 5·6-s − 7-s + 15·9-s + 5·11-s + 5·13-s + 14-s − 17-s − 15·18-s − 5·21-s − 5·22-s − 2·23-s − 25-s − 5·26-s + 35·27-s − 2·31-s + 25·33-s + 34-s + 25·39-s + 5·42-s + 2·46-s + 50-s − 5·51-s − 2·53-s − 35·54-s + 2·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{12} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{12} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(18.05972003\) |
\(L(\frac12)\) |
\(\approx\) |
\(18.05972003\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 7 | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 17 | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
good | 3 | \( ( 1 - T )^{6}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \) |
| 5 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \) |
| 11 | \( ( 1 - T )^{6}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \) |
| 13 | \( ( 1 - T )^{6}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \) |
| 19 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 23 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 29 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \) |
| 31 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 37 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \) |
| 41 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \) |
| 43 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \) |
| 47 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \) |
| 53 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 59 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \) |
| 61 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \) |
| 67 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 71 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 73 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \) |
| 79 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 83 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \) |
| 89 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 97 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
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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
−4.29502192060727304995159696964, −4.28736852378233598901268534655, −4.25594542847401194036956686213, −4.04689507810119786006619538020, −3.99028868104225092765684075007, −3.91918708090996316887330666189, −3.73745350884794681555926611202, −3.59836164524781691976066744660, −3.41001227877780259508319474166, −3.30978354497273307790501486054, −3.25706194755196163332829296173, −3.16864521892414047524088377917, −3.16384483852969260352581428576, −2.57563752139687124372397925501, −2.54755603560408072253024734036, −2.54178433631714906095271459887, −1.92063486695949818513799242389, −1.82250609616824249987916364299, −1.81553218313765028260279962571, −1.78167761231273215605079634324, −1.56155869353784588912339738545, −1.29223192304533435226780931431, −1.27612303701564556537178517596, −1.10940514060402664446714313459, −0.906870331352341292476389969740,
0.906870331352341292476389969740, 1.10940514060402664446714313459, 1.27612303701564556537178517596, 1.29223192304533435226780931431, 1.56155869353784588912339738545, 1.78167761231273215605079634324, 1.81553218313765028260279962571, 1.82250609616824249987916364299, 1.92063486695949818513799242389, 2.54178433631714906095271459887, 2.54755603560408072253024734036, 2.57563752139687124372397925501, 3.16384483852969260352581428576, 3.16864521892414047524088377917, 3.25706194755196163332829296173, 3.30978354497273307790501486054, 3.41001227877780259508319474166, 3.59836164524781691976066744660, 3.73745350884794681555926611202, 3.91918708090996316887330666189, 3.99028868104225092765684075007, 4.04689507810119786006619538020, 4.25594542847401194036956686213, 4.28736852378233598901268534655, 4.29502192060727304995159696964
Plot not available for L-functions of degree greater than 10.