| L(s) = 1 | − 8·2-s + 48·4-s + 18·5-s − 256·8-s − 144·10-s − 2·11-s − 288·13-s + 1.28e3·16-s + 1.53e3·17-s − 1.18e3·19-s + 864·20-s + 16·22-s − 3.39e3·23-s − 1.30e3·25-s + 2.30e3·26-s + 3.97e3·29-s − 7.59e3·31-s − 6.14e3·32-s − 1.22e4·34-s + 2.68e3·37-s + 9.50e3·38-s − 4.60e3·40-s + 3.66e4·41-s − 2.30e4·43-s − 96·44-s + 2.71e4·46-s + 864·47-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 0.321·5-s − 1.41·8-s − 0.455·10-s − 0.00498·11-s − 0.472·13-s + 5/4·16-s + 1.28·17-s − 0.754·19-s + 0.482·20-s + 0.00704·22-s − 1.33·23-s − 0.416·25-s + 0.668·26-s + 0.877·29-s − 1.41·31-s − 1.06·32-s − 1.81·34-s + 0.322·37-s + 1.06·38-s − 0.455·40-s + 3.40·41-s − 1.89·43-s − 0.00747·44-s + 1.88·46-s + 0.0570·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 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
−9.110987749020565277903405329118, −8.991401496919358282146114694328, −8.220400334419701025294000155163, −8.002081889761395316410637999759, −7.55430958542966741602841448909, −7.37080987351475880579049186904, −6.57017348092223294919284127802, −6.35577450239813307539667345171, −5.66512933122514901580679591304, −5.63181366803601933671415933532, −4.77707166905828990550874416558, −4.16936035130082708753465584189, −3.65836644967760464957873419020, −3.04556915047309937487484153527, −2.28413930294448249988341959805, −2.22497586143914281034170014326, −1.30001818799649595666258386704, −1.04420409080615409248609403174, 0, 0,
1.04420409080615409248609403174, 1.30001818799649595666258386704, 2.22497586143914281034170014326, 2.28413930294448249988341959805, 3.04556915047309937487484153527, 3.65836644967760464957873419020, 4.16936035130082708753465584189, 4.77707166905828990550874416558, 5.63181366803601933671415933532, 5.66512933122514901580679591304, 6.35577450239813307539667345171, 6.57017348092223294919284127802, 7.37080987351475880579049186904, 7.55430958542966741602841448909, 8.002081889761395316410637999759, 8.220400334419701025294000155163, 8.991401496919358282146114694328, 9.110987749020565277903405329118
