| L(s) = 1 | − 32·2-s + 768·4-s − 1.63e4·8-s + 4.28e3·9-s − 3.59e4·13-s + 3.27e5·16-s + 1.67e5·17-s − 1.37e5·18-s + 7.81e5·25-s + 1.14e6·26-s − 6.29e6·32-s − 5.34e6·34-s + 3.29e6·36-s + 1.10e7·49-s − 2.50e7·50-s − 2.75e7·52-s + 3.28e6·53-s + 1.17e8·64-s + 1.28e8·68-s − 7.02e7·72-s − 2.46e7·81-s + 2.50e8·89-s − 3.52e8·98-s + 6.00e8·100-s + 2.32e8·101-s + 5.88e8·104-s − 1.05e8·106-s + ⋯ |
| L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s + 0.653·9-s − 1.25·13-s + 5·16-s + 2·17-s − 1.30·18-s + 2·25-s + 2.51·26-s − 6·32-s − 4·34-s + 1.95·36-s + 1.91·49-s − 4·50-s − 3.77·52-s + 0.416·53-s + 7·64-s + 6·68-s − 2.61·72-s − 0.573·81-s + 3.99·89-s − 3.82·98-s + 6·100-s + 2.23·101-s + 5.02·104-s − 0.832·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4624 ^{s/2} \, \Gamma_{\C}(s+4)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.331463534\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.331463534\) |
| \(L(5)\) |
|
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 | $C_1$ | \( ( 1 + p^{4} T )^{2} \) |
| 17 | $C_1$ | \( ( 1 - p^{4} T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 4286 T^{2} + p^{16} T^{4} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 11031166 T^{2} + p^{16} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 233221438 T^{2} + p^{16} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 17954 T + p^{8} T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 120515130238 T^{2} + p^{16} T^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 1467981553154 T^{2} + p^{16} T^{4} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 1641314 T + p^{8} T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 1155787964270206 T^{2} + p^{16} T^{4} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 1984913441809154 T^{2} + p^{16} T^{4} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 125190718 T + p^{8} T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
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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
−13.08034541152767047229625012965, −12.45489236668265438476466302756, −12.10441323709458379854594777390, −11.67096278592703396993999660325, −10.77593615128436862150631242761, −10.19482595919980759191239182582, −10.17281454081007109358243997706, −9.298874033350059972159956422966, −8.930999530369151720780475977954, −8.149656972266624493494982641023, −7.43288765018142242700799668396, −7.31425089484848530249390263311, −6.52558512785894468041698983081, −5.70604783548131357660781481809, −4.94002235317504079045946031891, −3.50771038510215909259630800799, −2.79302393905603615274777571097, −2.01219034797475968050761970917, −1.03792250905949911191458595260, −0.64064755505708996072937560406,
0.64064755505708996072937560406, 1.03792250905949911191458595260, 2.01219034797475968050761970917, 2.79302393905603615274777571097, 3.50771038510215909259630800799, 4.94002235317504079045946031891, 5.70604783548131357660781481809, 6.52558512785894468041698983081, 7.31425089484848530249390263311, 7.43288765018142242700799668396, 8.149656972266624493494982641023, 8.930999530369151720780475977954, 9.298874033350059972159956422966, 10.17281454081007109358243997706, 10.19482595919980759191239182582, 10.77593615128436862150631242761, 11.67096278592703396993999660325, 12.10441323709458379854594777390, 12.45489236668265438476466302756, 13.08034541152767047229625012965
