| L(s) = 1 | + 2-s + 3-s + 6-s − 19-s − 29-s + 37-s − 38-s + 43-s + 3·49-s − 57-s − 58-s + 3·71-s + 73-s + 74-s − 79-s + 83-s + 86-s − 87-s − 89-s + 3·98-s − 101-s + 103-s − 6·107-s − 109-s + 111-s − 114-s + 3·121-s + ⋯ |
| L(s) = 1 | + 2-s + 3-s + 6-s − 19-s − 29-s + 37-s − 38-s + 43-s + 3·49-s − 57-s − 58-s + 3·71-s + 73-s + 74-s − 79-s + 83-s + 86-s − 87-s − 89-s + 3·98-s − 101-s + 103-s − 6·107-s − 109-s + 111-s − 114-s + 3·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 71^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 71^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.536996978\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.536996978\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 71 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 2 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 3 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 19 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 29 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 37 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 43 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 73 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 79 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 83 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 89 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
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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
−8.511963950730805483533931505158, −8.092999675564426862480072422022, −7.933467742952561953343259587929, −7.77319794355709658399529797980, −7.44634167506164238076235736069, −6.98849590131557118033005282891, −6.78231852590847581561049931265, −6.71976537633900584986348097572, −6.11764220346762438017732349981, −6.06543360272651378350888864909, −5.41491449903659888756647206503, −5.40012912993915675875647282975, −5.35883900664717898801667606591, −4.66445162884313331504616397613, −4.42320879553884568418502779802, −4.18730425919183222403248140492, −3.94666446874633105705542335049, −3.73038486337913326008770265232, −3.37656538460639035528715631921, −2.76957246544161024288190011392, −2.67976955216007079673286458685, −2.30469465717266151622483926916, −2.05808700313912555456813622741, −1.39693348684821972064541686038, −0.827836044704786324084222436229,
0.827836044704786324084222436229, 1.39693348684821972064541686038, 2.05808700313912555456813622741, 2.30469465717266151622483926916, 2.67976955216007079673286458685, 2.76957246544161024288190011392, 3.37656538460639035528715631921, 3.73038486337913326008770265232, 3.94666446874633105705542335049, 4.18730425919183222403248140492, 4.42320879553884568418502779802, 4.66445162884313331504616397613, 5.35883900664717898801667606591, 5.40012912993915675875647282975, 5.41491449903659888756647206503, 6.06543360272651378350888864909, 6.11764220346762438017732349981, 6.71976537633900584986348097572, 6.78231852590847581561049931265, 6.98849590131557118033005282891, 7.44634167506164238076235736069, 7.77319794355709658399529797980, 7.933467742952561953343259587929, 8.092999675564426862480072422022, 8.511963950730805483533931505158
