| L(s) = 1 | + 3·19-s + 30·37-s + 8·64-s + 21·73-s − 114·109-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 + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯ |
| L(s) = 1 | + 0.688·19-s + 4.93·37-s + 64-s + 2.45·73-s − 10.9·109-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 + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{36}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.377878341\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.377878341\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - p^{3} T^{6} + p^{6} T^{12} \) |
| 5 | \( 1 - p^{3} T^{6} + p^{6} T^{12} \) |
| 7 | \( ( 1 - 17 T^{3} + p^{3} T^{6} )( 1 + 37 T^{3} + p^{3} T^{6} ) \) |
| 11 | \( 1 - p^{3} T^{6} + p^{6} T^{12} \) |
| 13 | \( ( 1 - 89 T^{3} + p^{3} T^{6} )( 1 + 19 T^{3} + p^{3} T^{6} ) \) |
| 17 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{3} \) |
| 19 | \( ( 1 - 8 T + p T^{2} )^{3}( 1 + 7 T + p T^{2} )^{3} \) |
| 23 | \( 1 - p^{3} T^{6} + p^{6} T^{12} \) |
| 29 | \( 1 - p^{3} T^{6} + p^{6} T^{12} \) |
| 31 | \( ( 1 + 19 T^{3} + p^{3} T^{6} )( 1 + 289 T^{3} + p^{3} T^{6} ) \) |
| 37 | \( ( 1 - 11 T + p T^{2} )^{3}( 1 + T + p T^{2} )^{3} \) |
| 41 | \( 1 - p^{3} T^{6} + p^{6} T^{12} \) |
| 43 | \( ( 1 - 449 T^{3} + p^{3} T^{6} )( 1 - 71 T^{3} + p^{3} T^{6} ) \) |
| 47 | \( 1 - p^{3} T^{6} + p^{6} T^{12} \) |
| 53 | \( ( 1 + p T^{2} )^{6} \) |
| 59 | \( 1 - p^{3} T^{6} + p^{6} T^{12} \) |
| 61 | \( ( 1 - 719 T^{3} + p^{3} T^{6} )( 1 + 901 T^{3} + p^{3} T^{6} ) \) |
| 67 | \( ( 1 - 1007 T^{3} + p^{3} T^{6} )( 1 + 127 T^{3} + p^{3} T^{6} ) \) |
| 71 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{3} \) |
| 73 | \( ( 1 - 17 T + p T^{2} )^{3}( 1 + 10 T + p T^{2} )^{3} \) |
| 79 | \( ( 1 - 503 T^{3} + p^{3} T^{6} )( 1 + 1387 T^{3} + p^{3} T^{6} ) \) |
| 83 | \( 1 - p^{3} T^{6} + p^{6} T^{12} \) |
| 89 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{3} \) |
| 97 | \( ( 1 - 1853 T^{3} + p^{3} T^{6} )( 1 + 523 T^{3} + p^{3} T^{6} ) \) |
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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
−5.58629501134884344251630837262, −5.20201244562254585610395849962, −5.18741081629584981848113567264, −5.08023397356834089827238608801, −5.04902550145165744703040859174, −4.58885160991286545714637689621, −4.57828893963226823238894653893, −4.39226810190590639148762898278, −4.05019558106758557138363098820, −3.86852740133365889913321318730, −3.85779063245516187222085309511, −3.73357354768251862478401609262, −3.65892967745008029953096080381, −3.02029054062207989109891076167, −2.88971687724150994353734800156, −2.78895331319813385966778326699, −2.61517623242230174196041248963, −2.53487223280189715835326118140, −2.31871220956846850570265009291, −1.81409665349745421782827441751, −1.67746716554469764605378563990, −1.30855079418430445403231739178, −1.05832123159129290306861063711, −0.823563746339314533382540274738, −0.36754890654905086576193349311,
0.36754890654905086576193349311, 0.823563746339314533382540274738, 1.05832123159129290306861063711, 1.30855079418430445403231739178, 1.67746716554469764605378563990, 1.81409665349745421782827441751, 2.31871220956846850570265009291, 2.53487223280189715835326118140, 2.61517623242230174196041248963, 2.78895331319813385966778326699, 2.88971687724150994353734800156, 3.02029054062207989109891076167, 3.65892967745008029953096080381, 3.73357354768251862478401609262, 3.85779063245516187222085309511, 3.86852740133365889913321318730, 4.05019558106758557138363098820, 4.39226810190590639148762898278, 4.57828893963226823238894653893, 4.58885160991286545714637689621, 5.04902550145165744703040859174, 5.08023397356834089827238608801, 5.18741081629584981848113567264, 5.20201244562254585610395849962, 5.58629501134884344251630837262
Plot not available for L-functions of degree greater than 10.
