| L(s) = 1 | − 98·7-s − 100·11-s + 540·13-s − 1.15e3·17-s + 2.94e3·19-s − 2.68e3·23-s − 1.73e3·25-s − 996·29-s + 2.61e3·31-s − 3.49e3·37-s − 1.70e4·41-s + 1.36e4·43-s − 1.18e4·47-s + 7.20e3·49-s − 3.75e4·53-s + 6.32e3·59-s + 3.97e4·61-s − 2.73e4·67-s − 3.54e4·71-s + 6.41e4·73-s + 9.80e3·77-s − 1.10e5·79-s + 5.89e3·83-s − 1.67e5·89-s − 5.29e4·91-s − 1.18e4·97-s − 2.33e4·101-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 0.249·11-s + 0.886·13-s − 0.966·17-s + 1.87·19-s − 1.05·23-s − 0.554·25-s − 0.219·29-s + 0.488·31-s − 0.419·37-s − 1.58·41-s + 1.12·43-s − 0.781·47-s + 3/7·49-s − 1.83·53-s + 0.236·59-s + 1.36·61-s − 0.745·67-s − 0.834·71-s + 1.40·73-s + 0.188·77-s − 1.98·79-s + 0.0939·83-s − 2.23·89-s − 0.669·91-s − 0.128·97-s − 0.228·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 1734 T^{2} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 100 T + 320086 T^{2} + 100 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 540 T + 526462 T^{2} - 540 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 1152 T + 3130846 T^{2} + 1152 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2944 T + 5655798 T^{2} - 2944 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2684 T + 3830734 T^{2} + 2684 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 996 T + 37205902 T^{2} + 996 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2616 T + 41609662 T^{2} - 2616 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 3492 T + 72298414 T^{2} + 3492 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 17016 T + 272987742 T^{2} + 17016 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 13640 T + 279761990 T^{2} - 13640 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 11832 T + 202334814 T^{2} + 11832 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 37548 T + 17523254 p T^{2} + 37548 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6320 T + 780841414 T^{2} - 6320 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 39764 T + 1977385070 T^{2} - 39764 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 27376 T - 46144618 T^{2} + 27376 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 35460 T + 3145043502 T^{2} + 35460 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 64188 T + 4387114438 T^{2} - 64188 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 110088 T + 8439555358 T^{2} + 110088 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 5896 T + 7776870614 T^{2} - 5896 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 167160 T + 16792608382 T^{2} + 167160 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 11876 T + 16694014454 T^{2} + 11876 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−9.813705170254719233979561684052, −9.629314795905935918413780618966, −9.012239568418946961420434929235, −8.621178129407207304023885656272, −8.002332842402755477921973223396, −7.81610671662168246415477077947, −6.92472577299283338810298725808, −6.87545012454510705809558951853, −6.12087543698603210981862035714, −5.81480806161536052774846489447, −5.23072255860065621084378505294, −4.72789175974615907954250042752, −3.83068470321456031051844240210, −3.76708946813871199274633664951, −2.92364711999085192770274088194, −2.53853147203355179714018666535, −1.56547197652195493287159555486, −1.20356547762687194784091125185, 0, 0,
1.20356547762687194784091125185, 1.56547197652195493287159555486, 2.53853147203355179714018666535, 2.92364711999085192770274088194, 3.76708946813871199274633664951, 3.83068470321456031051844240210, 4.72789175974615907954250042752, 5.23072255860065621084378505294, 5.81480806161536052774846489447, 6.12087543698603210981862035714, 6.87545012454510705809558951853, 6.92472577299283338810298725808, 7.81610671662168246415477077947, 8.002332842402755477921973223396, 8.621178129407207304023885656272, 9.012239568418946961420434929235, 9.629314795905935918413780618966, 9.813705170254719233979561684052
