| L(s) = 1 | − 9·7-s − 8-s − 6·11-s − 8·27-s + 24·31-s − 12·37-s + 48·49-s + 9·56-s + 54·77-s + 18·83-s + 6·88-s + 18·103-s + 21·107-s + 84·113-s + 45·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 3.40·7-s − 0.353·8-s − 1.80·11-s − 1.53·27-s + 4.31·31-s − 1.97·37-s + 48/7·49-s + 1.20·56-s + 6.15·77-s + 1.97·83-s + 0.639·88-s + 1.77·103-s + 2.03·107-s + 7.90·113-s + 4.09·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6887221231\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6887221231\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T^{3} + T^{6} \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 8 T^{3} + 37 T^{6} + 8 p^{3} T^{9} + p^{6} T^{12} \) |
| 5 | \( 1 - 4 T^{3} - 109 T^{6} - 4 p^{3} T^{9} + p^{6} T^{12} \) |
| 7 | \( ( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{3} \) |
| 11 | \( ( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{3} \) |
| 13 | \( 1 + 38 T^{3} - 753 T^{6} + 38 p^{3} T^{9} + p^{6} T^{12} \) |
| 17 | \( 1 - 126 T^{3} + 10963 T^{6} - 126 p^{3} T^{9} + p^{6} T^{12} \) |
| 23 | \( 1 + 68 T^{3} - 7543 T^{6} + 68 p^{3} T^{9} + p^{6} T^{12} \) |
| 29 | \( 1 + 310 T^{3} + 71711 T^{6} + 310 p^{3} T^{9} + p^{6} T^{12} \) |
| 31 | \( ( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{3} \) |
| 37 | \( ( 1 + 2 T + p T^{2} )^{6} \) |
| 41 | \( 1 + 472 T^{3} + 153863 T^{6} + 472 p^{3} T^{9} + p^{6} T^{12} \) |
| 43 | \( 1 - 452 T^{3} + 124797 T^{6} - 452 p^{3} T^{9} + p^{6} T^{12} \) |
| 47 | \( 1 - 616 T^{3} + 275633 T^{6} - 616 p^{3} T^{9} + p^{6} T^{12} \) |
| 53 | \( 1 + 158 T^{3} - 123913 T^{6} + 158 p^{3} T^{9} + p^{6} T^{12} \) |
| 59 | \( 1 + 720 T^{3} + 313021 T^{6} + 720 p^{3} T^{9} + p^{6} T^{12} \) |
| 61 | \( 1 - 358 T^{3} - 98817 T^{6} - 358 p^{3} T^{9} + p^{6} T^{12} \) |
| 67 | \( 1 - 576 T^{3} + 31013 T^{6} - 576 p^{3} T^{9} + p^{6} T^{12} \) |
| 71 | \( 1 - 418 T^{3} - 183187 T^{6} - 418 p^{3} T^{9} + p^{6} T^{12} \) |
| 73 | \( 1 - 1242 T^{3} + 1153547 T^{6} - 1242 p^{3} T^{9} + p^{6} T^{12} \) |
| 79 | \( 1 + 1370 T^{3} + 1383861 T^{6} + 1370 p^{3} T^{9} + p^{6} T^{12} \) |
| 83 | \( ( 1 - 6 T - 47 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{3} \) |
| 89 | \( 1 - p^{3} T^{6} + p^{6} T^{12} \) |
| 97 | \( 1 + 574 T^{3} - 583197 T^{6} + 574 p^{3} T^{9} + p^{6} T^{12} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.58805224378258564511893935610, −5.50758334305204991516361997592, −5.45040992262970050496267113057, −4.91724642523211689240275911233, −4.89479701157154266189784871098, −4.62154498439553498294574565671, −4.46674233038011955136639474347, −4.46093337790663584358666792915, −4.31393606191331097456211681659, −3.96658836716027935821134805837, −3.49298788144690490746115928958, −3.40347053322413799094085903922, −3.30990014845757975455377241577, −3.21197856846635753144969647863, −3.20759091722044004082020246190, −3.15708211410575663912698735661, −2.42855879150439763908586196773, −2.37531334459146690581550722978, −2.32572427428437487014186549568, −2.13902452911847298068460175340, −1.90321475318731714571285585517, −1.21633162996391308057239264654, −0.75306377121620432118917381904, −0.70305837674325201585939812283, −0.23923187676243014600849423162,
0.23923187676243014600849423162, 0.70305837674325201585939812283, 0.75306377121620432118917381904, 1.21633162996391308057239264654, 1.90321475318731714571285585517, 2.13902452911847298068460175340, 2.32572427428437487014186549568, 2.37531334459146690581550722978, 2.42855879150439763908586196773, 3.15708211410575663912698735661, 3.20759091722044004082020246190, 3.21197856846635753144969647863, 3.30990014845757975455377241577, 3.40347053322413799094085903922, 3.49298788144690490746115928958, 3.96658836716027935821134805837, 4.31393606191331097456211681659, 4.46093337790663584358666792915, 4.46674233038011955136639474347, 4.62154498439553498294574565671, 4.89479701157154266189784871098, 4.91724642523211689240275911233, 5.45040992262970050496267113057, 5.50758334305204991516361997592, 5.58805224378258564511893935610
Plot not available for L-functions of degree greater than 10.
