| L(s) = 1 | + 729·3-s + 7.68e4·7-s + 3.54e5·9-s − 1.12e7·19-s + 5.60e7·21-s + 4.88e7·25-s + 1.29e8·27-s + 5.46e8·31-s + 1.19e8·37-s + 4.37e8·43-s + 3.93e9·49-s − 8.21e9·57-s − 2.15e10·61-s + 2.72e10·63-s − 5.95e9·67-s + 5.51e10·73-s + 3.55e10·75-s + 5.42e10·79-s + 3.13e10·81-s + 3.98e11·93-s + 8.41e10·103-s + 2.57e9·109-s + 8.70e10·111-s − 2.85e11·121-s + 127-s + 3.19e11·129-s + 131-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 1.72·7-s + 2·9-s − 1.04·19-s + 2.99·21-s + 25-s + 1.73·27-s + 3.42·31-s + 0.283·37-s + 0.454·43-s + 1.98·49-s − 1.80·57-s − 3.26·61-s + 3.45·63-s − 0.538·67-s + 3.11·73-s + 1.73·75-s + 1.98·79-s + 81-s + 5.93·93-s + 0.714·103-s + 0.0160·109-s + 0.490·111-s − 121-s + 0.786·129-s − 1.80·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+11/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(10.90263385\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.90263385\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - p^{6} T + p^{11} T^{2} \) |
| 7 | $C_2$ | \( 1 - 76885 T + p^{11} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p^{11} T^{2} + p^{22} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + p^{11} T^{2} + p^{22} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2248615 T + p^{11} T^{2} )( 1 + 2248615 T + p^{11} T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - p^{11} T^{2} + p^{22} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4655368 T + p^{11} T^{2} )( 1 + 15926533 T + p^{11} T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + p^{11} T^{2} + p^{22} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p^{11} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 296476943 T + p^{11} T^{2} )( 1 - 249734764 T + p^{11} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 782919730 T + p^{11} T^{2} )( 1 + 663545123 T + p^{11} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p^{11} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 218924719 T + p^{11} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p^{11} T^{2} + p^{22} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + p^{11} T^{2} + p^{22} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p^{11} T^{2} + p^{22} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 8546352539 T + p^{11} T^{2} )( 1 + 12977292913 T + p^{11} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 15458751248 T + p^{11} T^{2} )( 1 + 21410042863 T + p^{11} T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - p^{11} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 35338681231 T + p^{11} T^{2} )( 1 - 19805520230 T + p^{11} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 32885832404 T + p^{11} T^{2} )( 1 - 21410392133 T + p^{11} T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p^{11} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p^{11} T^{2} + p^{22} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 112637211442 T + p^{11} T^{2} )( 1 + 112637211442 T + p^{11} T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.19939531135803608820017532140, −11.99530083699249016661076669644, −10.94959930685214184721803372894, −10.75048717822967744100274847660, −10.03842993332988248886229455506, −9.383785206821784841484280762544, −8.802277371994578979957015976349, −8.339808346975697969400086992909, −7.980700800854387752275336793228, −7.59348123397043305801973336736, −6.70654117064118631774547790776, −6.14140949834141801708416814708, −4.82905951358509689272044913458, −4.69554157909811064437304811277, −4.03725895555052673700095112863, −3.14195852561976817387749498775, −2.51878079443057464172103616509, −2.04907110796179886054655932273, −1.29760995943721778137097901463, −0.77120180215953608406599383453,
0.77120180215953608406599383453, 1.29760995943721778137097901463, 2.04907110796179886054655932273, 2.51878079443057464172103616509, 3.14195852561976817387749498775, 4.03725895555052673700095112863, 4.69554157909811064437304811277, 4.82905951358509689272044913458, 6.14140949834141801708416814708, 6.70654117064118631774547790776, 7.59348123397043305801973336736, 7.980700800854387752275336793228, 8.339808346975697969400086992909, 8.802277371994578979957015976349, 9.383785206821784841484280762544, 10.03842993332988248886229455506, 10.75048717822967744100274847660, 10.94959930685214184721803372894, 11.99530083699249016661076669644, 12.19939531135803608820017532140
