| L(s) = 1 | − 6·5-s + 8·7-s + 66·11-s − 2·13-s + 34·17-s + 26·19-s − 198·23-s + 65·25-s − 444·29-s − 532·31-s − 48·35-s + 88·37-s − 570·41-s + 182·43-s + 420·47-s − 350·49-s + 300·53-s − 396·55-s + 444·59-s + 400·61-s + 12·65-s + 968·67-s − 1.84e3·71-s + 868·73-s + 528·77-s − 508·79-s + 996·83-s + ⋯ |
| L(s) = 1 | − 0.536·5-s + 0.431·7-s + 1.80·11-s − 0.0426·13-s + 0.485·17-s + 0.313·19-s − 1.79·23-s + 0.519·25-s − 2.84·29-s − 3.08·31-s − 0.231·35-s + 0.391·37-s − 2.17·41-s + 0.645·43-s + 1.30·47-s − 1.02·49-s + 0.777·53-s − 0.970·55-s + 0.979·59-s + 0.839·61-s + 0.0228·65-s + 1.76·67-s − 3.08·71-s + 1.39·73-s + 0.781·77-s − 0.723·79-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5992704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5992704 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 17 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 6 T - 29 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 8 T + 414 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 6 p T + 3463 T^{2} - 6 p^{4} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 3243 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 26 T + 9279 T^{2} - 26 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 198 T + 33847 T^{2} + 198 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 444 T + 93454 T^{2} + 444 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 532 T + 129186 T^{2} + 532 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 88 T + 29514 T^{2} - 88 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 570 T + 195739 T^{2} + 570 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 182 T + 162687 T^{2} - 182 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 420 T + 249154 T^{2} - 420 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 300 T + 43486 T^{2} - 300 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 444 T + 294154 T^{2} - 444 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 400 T + 55914 T^{2} - 400 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 968 T + 742470 T^{2} - 968 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1848 T + 1513150 T^{2} + 1848 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 868 T + 963798 T^{2} - 868 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 254 T + p^{3} T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 12 p T + 1388986 T^{2} - 12 p^{4} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 288 T + 287602 T^{2} - 288 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 1024 T + 2038818 T^{2} - 1024 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.422024823698766335151986103207, −8.023341385105861905415470764365, −7.52713778546927101666072675140, −7.23949633469583897286916265738, −7.08718702090237851473897582920, −6.50099771903652399390639490435, −5.97982973241466508973774759988, −5.68761387635086453795805780521, −5.29799673283590802120098783548, −4.92971069810404900034389896748, −4.07073892194276614199972355869, −3.95667119618461294722235587627, −3.58715516198928848896644792311, −3.43409797141716385570576574969, −2.28623923379466605216691623557, −2.03194944621552592447389685025, −1.49271219230403387270106696696, −1.07170118466239084073197173761, 0, 0,
1.07170118466239084073197173761, 1.49271219230403387270106696696, 2.03194944621552592447389685025, 2.28623923379466605216691623557, 3.43409797141716385570576574969, 3.58715516198928848896644792311, 3.95667119618461294722235587627, 4.07073892194276614199972355869, 4.92971069810404900034389896748, 5.29799673283590802120098783548, 5.68761387635086453795805780521, 5.97982973241466508973774759988, 6.50099771903652399390639490435, 7.08718702090237851473897582920, 7.23949633469583897286916265738, 7.52713778546927101666072675140, 8.023341385105861905415470764365, 8.422024823698766335151986103207
