| L(s) = 1 | − 126·3-s − 5.57e3·7-s + 9.31e3·9-s + 2.63e4·13-s + 2.88e5·19-s + 7.02e5·21-s + 4.48e5·25-s − 3.47e5·27-s − 1.45e6·31-s + 3.92e6·37-s − 3.31e6·39-s − 1.56e5·43-s + 1.17e7·49-s − 3.62e7·57-s − 3.51e7·61-s − 5.19e7·63-s − 3.42e7·67-s + 5.62e7·73-s − 5.64e7·75-s − 1.83e7·79-s − 1.73e7·81-s − 1.46e8·91-s + 1.83e8·93-s − 2.57e8·97-s − 5.43e8·109-s − 4.95e8·111-s + 2.44e8·117-s + ⋯ |
| L(s) = 1 | − 1.55·3-s − 2.32·7-s + 1.41·9-s + 0.920·13-s + 2.20·19-s + 3.60·21-s + 1.14·25-s − 0.652·27-s − 1.57·31-s + 2.09·37-s − 1.43·39-s − 0.0457·43-s + 2.03·49-s − 3.43·57-s − 2.53·61-s − 3.29·63-s − 1.70·67-s + 1.98·73-s − 1.78·75-s − 0.471·79-s − 0.404·81-s − 2.13·91-s + 2.45·93-s − 2.90·97-s − 3.85·109-s − 3.26·111-s + 1.30·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s+4)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.06190869019\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06190869019\) |
| \(L(5)\) |
|
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 | $C_2$ | \( 1 + 14 p^{2} T + p^{8} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 448322 T^{2} + p^{16} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 398 p T + p^{8} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 74615230 T^{2} + p^{16} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 13150 T + p^{8} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 9544036610 T^{2} + p^{16} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 144002 T + p^{8} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 154186508930 T^{2} + p^{16} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 606859926722 T^{2} + p^{16} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 728738 T + p^{8} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 1964446 T + p^{8} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 14997407035010 T^{2} + p^{16} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 78142 T + p^{8} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 35234322443522 T^{2} + p^{16} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 124246846237250 T^{2} + p^{16} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 268618401162050 T^{2} + p^{16} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 17578274 T + p^{8} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 17136766 T + p^{8} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 621182343784322 T^{2} + p^{16} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 28139330 T + p^{8} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 9182498 T + p^{8} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 3083295701563966 T^{2} + p^{16} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 1271157775602050 T^{2} + p^{16} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 128722558 T + p^{8} T^{2} )^{2} \) |
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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
−11.27842132424535676489518340011, −10.74702631597041894625329361422, −10.55797485071300290038199046690, −9.649564183896836964609948613275, −9.500713611456830895797617202516, −9.226629261037411717674908961276, −8.265564573597418229813744956856, −7.43297184495942713747825060255, −7.12331809745398978894000531445, −6.41154097679017247876098822429, −6.23277526712282718603787827700, −5.66459456218300965103038959200, −5.23191954487186675238405664344, −4.44488754505901311002922659413, −3.69446511946735511213971824553, −3.16310398887605375498984123926, −2.73354332234528429171769849027, −1.26385910026616628852121778392, −1.04908825119493175411588131662, −0.079939641677210007811450181877,
0.079939641677210007811450181877, 1.04908825119493175411588131662, 1.26385910026616628852121778392, 2.73354332234528429171769849027, 3.16310398887605375498984123926, 3.69446511946735511213971824553, 4.44488754505901311002922659413, 5.23191954487186675238405664344, 5.66459456218300965103038959200, 6.23277526712282718603787827700, 6.41154097679017247876098822429, 7.12331809745398978894000531445, 7.43297184495942713747825060255, 8.265564573597418229813744956856, 9.226629261037411717674908961276, 9.500713611456830895797617202516, 9.649564183896836964609948613275, 10.55797485071300290038199046690, 10.74702631597041894625329361422, 11.27842132424535676489518340011
