| L(s) = 1 | − 48·5-s − 98·7-s − 368·11-s − 156·13-s + 3.31e3·17-s + 3.73e3·19-s − 1.55e3·23-s − 822·25-s − 1.72e3·29-s − 3.62e3·31-s + 4.70e3·35-s + 6.99e3·37-s − 2.61e4·41-s − 3.01e4·43-s + 1.14e4·47-s + 7.20e3·49-s + 1.73e4·53-s + 1.76e4·55-s − 1.50e4·59-s + 3.55e4·61-s + 7.48e3·65-s − 7.05e4·67-s − 4.07e4·71-s − 5.38e4·73-s + 3.60e4·77-s − 9.74e3·79-s − 3.13e4·83-s + ⋯ |
| L(s) = 1 | − 0.858·5-s − 0.755·7-s − 0.916·11-s − 0.256·13-s + 2.77·17-s + 2.37·19-s − 0.611·23-s − 0.263·25-s − 0.381·29-s − 0.677·31-s + 0.649·35-s + 0.840·37-s − 2.43·41-s − 2.48·43-s + 0.754·47-s + 3/7·49-s + 0.849·53-s + 0.787·55-s − 0.561·59-s + 1.22·61-s + 0.219·65-s − 1.91·67-s − 0.959·71-s − 1.18·73-s + 0.693·77-s − 0.175·79-s − 0.499·83-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 | $D_{4}$ | \( 1 + 48 T + 3126 T^{2} + 48 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 368 T + 153346 T^{2} + 368 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 12 p T - 198530 T^{2} + 12 p^{6} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 3312 T + 5420878 T^{2} - 3312 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 3736 T + 8100630 T^{2} - 3736 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 1552 T + 13432090 T^{2} + 1552 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 1728 T + 36312922 T^{2} + 1728 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 3624 T + 60503758 T^{2} + 3624 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 6996 T + 149067406 T^{2} - 6996 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 26160 T + 385589214 T^{2} + 26160 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 30184 T + 521633798 T^{2} + 30184 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 11424 T + 490819086 T^{2} - 11424 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 17376 T + 113903098 T^{2} - 17376 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 15008 T + 1001295814 T^{2} + 15008 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 35564 T + 1803486974 T^{2} - 35564 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 70504 T + 3904156406 T^{2} + 70504 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 40752 T + 3146114778 T^{2} + 40752 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 53892 T + 4664755414 T^{2} + 53892 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 9744 T + 5082885982 T^{2} + 9744 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 31360 T - 880376314 T^{2} + 31360 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 35952 T + 5931688606 T^{2} + 35952 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 66652 T - 1905212410 T^{2} - 66652 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−9.844306537761675583539240417421, −9.832143054581215400771340419018, −9.015874147173357020620724691811, −8.472112837770220845090996379976, −7.897845138066902062837443685705, −7.71346087845567337359696755576, −7.23457011871703284302164576826, −7.00261173633730879587092009520, −5.85805154960582377956129189806, −5.84959578320191004698779380989, −5.10945048162163181854997521619, −4.94493222122853915419641918525, −3.78453165308380395136656504340, −3.54818657387661789057257202313, −3.14583292543999744662694994568, −2.62921914351541606094174639534, −1.47511234982050315131037759899, −1.13370315401430579337387613662, 0, 0,
1.13370315401430579337387613662, 1.47511234982050315131037759899, 2.62921914351541606094174639534, 3.14583292543999744662694994568, 3.54818657387661789057257202313, 3.78453165308380395136656504340, 4.94493222122853915419641918525, 5.10945048162163181854997521619, 5.84959578320191004698779380989, 5.85805154960582377956129189806, 7.00261173633730879587092009520, 7.23457011871703284302164576826, 7.71346087845567337359696755576, 7.897845138066902062837443685705, 8.472112837770220845090996379976, 9.015874147173357020620724691811, 9.832143054581215400771340419018, 9.844306537761675583539240417421
