| L(s) = 1 | − 1.77e5·9-s + 2.68e5·13-s + 9.76e7·25-s − 1.56e9·37-s − 3.95e9·49-s + 8.86e9·61-s − 3.96e10·73-s + 3.13e10·81-s + 2.25e11·97-s + 5.59e11·109-s − 4.76e10·117-s − 5.70e11·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 3.53e12·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 9-s + 0.200·13-s + 2·25-s − 3.71·37-s − 1.99·49-s + 1.34·61-s − 2.23·73-s + 81-s + 2.66·97-s + 3.48·109-s − 0.200·117-s − 2·121-s − 1.96·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+11/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.288544169\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.288544169\) |
| \(L(\frac{13}{2})\) |
|
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 | $C_2$ | \( 1 + p^{11} T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - p^{11} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 268 T + p^{11} T^{2} )( 1 + 268 T + p^{11} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p^{11} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 134374 T + p^{11} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p^{11} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4655368 T + p^{11} T^{2} )( 1 + 4655368 T + p^{11} T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p^{11} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p^{11} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 249734764 T + p^{11} T^{2} )( 1 + 249734764 T + p^{11} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 782919730 T + p^{11} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p^{11} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 1549433416 T + p^{11} T^{2} )( 1 + 1549433416 T + p^{11} T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p^{11} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p^{11} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p^{11} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4430940374 T + p^{11} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 15458751248 T + p^{11} T^{2} )( 1 + 15458751248 T + p^{11} T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p^{11} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 19805520230 T + p^{11} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 32885832404 T + p^{11} T^{2} )( 1 + 32885832404 T + p^{11} T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p^{11} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p^{11} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 112637211442 T + p^{11} T^{2} )^{2} \) |
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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
−14.29671864628947633858229058998, −12.87521275255102729806610637556, −12.47813935351785090017610228320, −11.66708235985864071320632138206, −11.35296574351557339054072021748, −10.43927424728340268983975518862, −10.28410732579485460342291751191, −9.132118983095537267818292112196, −8.723414389043755783880375305942, −8.299442781636224205641585700130, −7.29587469656523357981358538701, −6.75026309462993928986246356180, −6.03135686040283863825138800760, −5.18181927638631294327335429953, −4.77753538574004750646609611650, −3.49910322979076867976758297198, −3.14609592369397111654046175982, −2.14691759097885883820941383215, −1.31274633033927892913034908260, −0.34318038960425904470324269802,
0.34318038960425904470324269802, 1.31274633033927892913034908260, 2.14691759097885883820941383215, 3.14609592369397111654046175982, 3.49910322979076867976758297198, 4.77753538574004750646609611650, 5.18181927638631294327335429953, 6.03135686040283863825138800760, 6.75026309462993928986246356180, 7.29587469656523357981358538701, 8.299442781636224205641585700130, 8.723414389043755783880375305942, 9.132118983095537267818292112196, 10.28410732579485460342291751191, 10.43927424728340268983975518862, 11.35296574351557339054072021748, 11.66708235985864071320632138206, 12.47813935351785090017610228320, 12.87521275255102729806610637556, 14.29671864628947633858229058998
