| L(s) = 1 | + 8·2-s − 18·3-s + 48·4-s − 18·5-s − 144·6-s + 256·8-s + 243·9-s − 144·10-s + 2·11-s − 864·12-s − 288·13-s + 324·15-s + 1.28e3·16-s − 1.53e3·17-s + 1.94e3·18-s − 1.18e3·19-s − 864·20-s + 16·22-s + 3.39e3·23-s − 4.60e3·24-s − 1.30e3·25-s − 2.30e3·26-s − 2.91e3·27-s − 3.97e3·29-s + 2.59e3·30-s − 7.59e3·31-s + 6.14e3·32-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 3/2·4-s − 0.321·5-s − 1.63·6-s + 1.41·8-s + 9-s − 0.455·10-s + 0.00498·11-s − 1.73·12-s − 0.472·13-s + 0.371·15-s + 5/4·16-s − 1.28·17-s + 1.41·18-s − 0.754·19-s − 0.482·20-s + 0.00704·22-s + 1.33·23-s − 1.63·24-s − 0.416·25-s − 0.668·26-s − 0.769·27-s − 0.877·29-s + 0.525·30-s − 1.41·31-s + 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 + 18 T + 1626 T^{2} + 18 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T - 59002 T^{2} - 2 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 288 T + 688042 T^{2} + 288 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 90 p T + 2366314 T^{2} + 90 p^{6} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 1188 T + 4834534 T^{2} + 1188 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 3390 T + 6218086 T^{2} - 3390 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3976 T + 31254662 T^{2} + 3976 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 7596 T + 61727326 T^{2} + 7596 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2688 T + 126774470 T^{2} - 2688 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 36630 T + 559995322 T^{2} + 36630 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 23032 T + 329072262 T^{2} + 23032 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 864 T + 46643358 T^{2} + 864 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 32920 T + 983844566 T^{2} + 32920 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 26712 T + 697343334 T^{2} - 26712 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 20412 T + 1230091258 T^{2} + 20412 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 36172 T + 33148290 p T^{2} + 36172 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 73706 T + 3356433686 T^{2} - 73706 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 74772 T + 3993671602 T^{2} - 74772 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 23116 T + 6249589662 T^{2} + 23116 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 147816 T + 13316083030 T^{2} + 147816 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 164646 T + 17878567722 T^{2} + 164646 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 162036 T + 20528488258 T^{2} + 162036 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−10.94051470496383352451219154701, −10.62306998003108148428554252806, −9.810306264878780583963266579585, −9.603927862172053627915586273507, −8.553666883184945449629858081451, −8.362188324449009716368627233228, −7.29526775415216150857973534705, −7.14889230383366722223268502247, −6.50884808899763086003673909495, −6.35243652157073466088126387616, −5.31144498956551657011629354982, −5.24575952664842368545525495579, −4.71043719111555090532257570228, −4.06735968326835452851113059023, −3.56773574337593234013412674276, −2.85417082178007813404195012162, −1.88309949483789137823819999482, −1.52483767391626227590160960001, 0, 0,
1.52483767391626227590160960001, 1.88309949483789137823819999482, 2.85417082178007813404195012162, 3.56773574337593234013412674276, 4.06735968326835452851113059023, 4.71043719111555090532257570228, 5.24575952664842368545525495579, 5.31144498956551657011629354982, 6.35243652157073466088126387616, 6.50884808899763086003673909495, 7.14889230383366722223268502247, 7.29526775415216150857973534705, 8.362188324449009716368627233228, 8.553666883184945449629858081451, 9.603927862172053627915586273507, 9.810306264878780583963266579585, 10.62306998003108148428554252806, 10.94051470496383352451219154701
