| L(s) = 1 | − 3-s − 5-s − 3·7-s + 3·9-s + 2·11-s + 5·13-s + 15-s − 10·17-s − 14·19-s + 3·21-s − 5·23-s + 2·25-s − 8·27-s − 3·29-s − 7·31-s − 2·33-s + 3·35-s − 12·37-s − 5·39-s + 12·41-s + 8·43-s − 3·45-s − 3·47-s + 8·49-s + 10·51-s + 20·53-s − 2·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.13·7-s + 9-s + 0.603·11-s + 1.38·13-s + 0.258·15-s − 2.42·17-s − 3.21·19-s + 0.654·21-s − 1.04·23-s + 2/5·25-s − 1.53·27-s − 0.557·29-s − 1.25·31-s − 0.348·33-s + 0.507·35-s − 1.97·37-s − 0.800·39-s + 1.87·41-s + 1.21·43-s − 0.447·45-s − 0.437·47-s + 8/7·49-s + 1.40·51-s + 2.74·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7977493164\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7977493164\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 + T - T^{2} - 8 T^{3} - 26 T^{4} - 8 p T^{5} - p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + 3 T + T^{2} - 18 T^{3} - 48 T^{4} - 18 p T^{5} + p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 5 T + T^{2} + 10 T^{3} + 82 T^{4} + 10 p T^{5} + p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 5 T + 32 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 7 T + 42 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 5 T - 19 T^{2} - 10 T^{3} + 832 T^{4} - 10 p T^{5} - 19 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 3 T - 43 T^{2} - 18 T^{3} + 1602 T^{4} - 18 p T^{5} - 43 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 7 T - 17 T^{2} + 28 T^{3} + 1876 T^{4} + 28 p T^{5} - 17 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 12 T + 59 T^{2} - 36 T^{3} - 360 T^{4} - 36 p T^{5} + 59 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 8 T - 5 T^{2} + 136 T^{3} + 160 T^{4} + 136 p T^{5} - 5 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 3 T - 79 T^{2} - 18 T^{3} + 5112 T^{4} - 18 p T^{5} - 79 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 10 T + 98 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 7 T - 10 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 + T - 113 T^{2} - 8 T^{3} + 9214 T^{4} - 8 p T^{5} - 113 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 4 T - 89 T^{2} + 116 T^{3} + 5464 T^{4} + 116 p T^{5} - 89 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 73 | $D_{4}$ | \( ( 1 + 7 T + 84 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 7 T - 113 T^{2} - 28 T^{3} + 16132 T^{4} - 28 p T^{5} - 113 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 25 T + 311 T^{2} - 3700 T^{3} + 39832 T^{4} - 3700 p T^{5} + 311 p^{2} T^{6} - 25 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 97 | $D_4\times C_2$ | \( 1 - 8 T - 113 T^{2} + 136 T^{3} + 15712 T^{4} + 136 p T^{5} - 113 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.78779782808171112512966871612, −7.22715514309356201752284757145, −7.21759917664092901124230745254, −6.95574285787064920799539087091, −6.80434005018416036609538601604, −6.58288246111339101907755245212, −6.25026889860613399660411280256, −5.99034159174410110023806969547, −5.98737232361202366826426560155, −5.71053505882593086017896797736, −5.46898259604252434834698099105, −4.79386549558208309588944130720, −4.71593546106292449136769565214, −4.34377646760899051440474906343, −4.11006399251522869937958301214, −3.96687817422801291273168861651, −3.86147187858533228258776729899, −3.35449600559463227719451255998, −3.31885878041627822419052565825, −2.34716446829473048557328915308, −2.17532591036492324447077593656, −2.10789157058983036020927388752, −1.76210234058830833101696524024, −0.810704621221195081761531485835, −0.36498634137254296746321726871,
0.36498634137254296746321726871, 0.810704621221195081761531485835, 1.76210234058830833101696524024, 2.10789157058983036020927388752, 2.17532591036492324447077593656, 2.34716446829473048557328915308, 3.31885878041627822419052565825, 3.35449600559463227719451255998, 3.86147187858533228258776729899, 3.96687817422801291273168861651, 4.11006399251522869937958301214, 4.34377646760899051440474906343, 4.71593546106292449136769565214, 4.79386549558208309588944130720, 5.46898259604252434834698099105, 5.71053505882593086017896797736, 5.98737232361202366826426560155, 5.99034159174410110023806969547, 6.25026889860613399660411280256, 6.58288246111339101907755245212, 6.80434005018416036609538601604, 6.95574285787064920799539087091, 7.21759917664092901124230745254, 7.22715514309356201752284757145, 7.78779782808171112512966871612
