L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 9-s − 10-s − 12-s + 15-s + 18-s − 19-s − 20-s + 25-s − 2·29-s + 30-s + 36-s + 37-s − 38-s − 2·43-s − 45-s + 50-s + 57-s − 2·58-s + 60-s + 6·71-s − 73-s + 74-s + ⋯ |
L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 9-s − 10-s − 12-s + 15-s + 18-s − 19-s − 20-s + 25-s − 2·29-s + 30-s + 36-s + 37-s − 38-s − 2·43-s − 45-s + 50-s + 57-s − 2·58-s + 60-s + 6·71-s − 73-s + 74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 71^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 71^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7965861416\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7965861416\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 71 | \( ( 1 - T )^{6} \) |
good | 2 | \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \) |
| 3 | \( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} \) |
| 5 | \( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} \) |
| 11 | \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \) |
| 13 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 17 | \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \) |
| 19 | \( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} \) |
| 23 | \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \) |
| 29 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 31 | \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \) |
| 37 | \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \) |
| 41 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 43 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 47 | \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \) |
| 53 | \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \) |
| 59 | \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \) |
| 61 | \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \) |
| 67 | \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \) |
| 73 | \( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} \) |
| 79 | \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \) |
| 83 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \) |
| 89 | \( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} \) |
| 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
−4.77376104869833063517286069343, −4.41498166165428092065482233790, −4.24809350178725615926188140225, −4.11970602463165251479688294919, −4.09958803401910367310204421325, −3.99039118876879559910045785496, −3.82550209665770489410397967094, −3.78589521685747149767288877673, −3.41311547954130792997050937537, −3.36135038406612570470186533712, −3.15029593883712429225745145638, −3.10085303291083657675933740530, −3.05734201112373189411048064456, −2.68509212904727638370804498193, −2.43399814268385564900794662257, −2.38272264615621835841131478667, −2.27431945835833468254751870976, −2.14581270282858342965753304725, −1.67694288617284494050679342443, −1.67118954799745648791099168456, −1.61377295561557172607681498835, −1.12835489749180291546672716273, −0.890568903070248866275676630216, −0.884571948817849579735741265292, −0.25655402626029747017831756830,
0.25655402626029747017831756830, 0.884571948817849579735741265292, 0.890568903070248866275676630216, 1.12835489749180291546672716273, 1.61377295561557172607681498835, 1.67118954799745648791099168456, 1.67694288617284494050679342443, 2.14581270282858342965753304725, 2.27431945835833468254751870976, 2.38272264615621835841131478667, 2.43399814268385564900794662257, 2.68509212904727638370804498193, 3.05734201112373189411048064456, 3.10085303291083657675933740530, 3.15029593883712429225745145638, 3.36135038406612570470186533712, 3.41311547954130792997050937537, 3.78589521685747149767288877673, 3.82550209665770489410397967094, 3.99039118876879559910045785496, 4.09958803401910367310204421325, 4.11970602463165251479688294919, 4.24809350178725615926188140225, 4.41498166165428092065482233790, 4.77376104869833063517286069343
Plot not available for L-functions of degree greater than 10.