| L(s) = 1 | − 32·2-s + 768·4-s − 1.63e4·8-s − 1.31e4·9-s + 956·13-s + 3.27e5·16-s − 6.33e4·17-s + 4.19e5·18-s + 3.29e5·25-s − 3.05e4·26-s − 6.29e6·32-s + 2.02e6·34-s − 1.00e7·36-s − 1.15e7·49-s − 1.05e7·50-s + 7.34e5·52-s − 1.92e7·53-s + 1.17e8·64-s − 4.86e7·68-s + 2.14e8·72-s + 1.29e8·81-s + 6.05e7·89-s + 3.68e8·98-s + 2.53e8·100-s − 2.90e8·101-s − 1.56e7·104-s + 6.15e8·106-s + ⋯ |
| L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s − 2·9-s + 0.0334·13-s + 5·16-s − 0.758·17-s + 4·18-s + 0.843·25-s − 0.0669·26-s − 6·32-s + 1.51·34-s − 6·36-s − 2·49-s − 1.68·50-s + 0.100·52-s − 2.43·53-s + 7·64-s − 2.27·68-s + 8·72-s + 3·81-s + 0.964·89-s + 4·98-s + 2.53·100-s − 2.79·101-s − 0.133·104-s + 4.87·106-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
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.04874554120\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04874554120\) |
| \(L(5)\) |
|
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 | $C_2$ | \( 1 + 63358 T + p^{8} T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p^{8} T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 1054 T + p^{8} T^{2} )( 1 + 1054 T + p^{8} T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p^{8} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p^{8} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 478 T + p^{8} T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p^{8} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 1407838 T + p^{8} T^{2} )( 1 + 1407838 T + p^{8} T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p^{8} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 925922 T + p^{8} T^{2} )( 1 + 925922 T + p^{8} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 3577922 T + p^{8} T^{2} )( 1 + 3577922 T + p^{8} T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + 9620638 T + p^{8} T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 20722082 T + p^{8} T^{2} )( 1 + 20722082 T + p^{8} T^{2} ) \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p^{8} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 54717118 T + p^{8} T^{2} )( 1 + 54717118 T + p^{8} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p^{8} T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 30265918 T + p^{8} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 173379838 T + p^{8} T^{2} )( 1 + 173379838 T + p^{8} T^{2} ) \) |
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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
−13.30741311945196445296271825831, −12.45492775979276220020283625355, −12.01879218338887643094920127351, −11.23058128738365636388412158140, −11.14223198155210443576095562996, −10.64579290541682406031929276760, −9.755006686381388233039321597488, −9.259052958656908872868312694446, −8.817784329779962117623562722649, −8.135207196498449265624796129018, −7.976318226377766267243741049673, −6.90935839509991216858677360389, −6.40751947519907574517338824654, −5.85605240136429091046807926977, −4.99647168717644692048001800292, −3.34559666383168507136103396919, −2.84883724868557697025648468163, −2.10379894229375742357572140505, −1.17861455349654095078947421113, −0.10476209224489035537393831253,
0.10476209224489035537393831253, 1.17861455349654095078947421113, 2.10379894229375742357572140505, 2.84883724868557697025648468163, 3.34559666383168507136103396919, 4.99647168717644692048001800292, 5.85605240136429091046807926977, 6.40751947519907574517338824654, 6.90935839509991216858677360389, 7.976318226377766267243741049673, 8.135207196498449265624796129018, 8.817784329779962117623562722649, 9.259052958656908872868312694446, 9.755006686381388233039321597488, 10.64579290541682406031929276760, 11.14223198155210443576095562996, 11.23058128738365636388412158140, 12.01879218338887643094920127351, 12.45492775979276220020283625355, 13.30741311945196445296271825831
