| L(s) = 1 | − 8·5-s − 24·7-s + 2·11-s − 32·13-s − 56·17-s + 184·19-s − 92·23-s + 95·25-s + 336·29-s − 376·31-s + 192·35-s − 348·37-s − 312·41-s + 80·43-s + 228·47-s + 43·49-s − 152·53-s − 16·55-s + 680·59-s + 112·61-s + 256·65-s + 352·67-s − 1.81e3·71-s − 574·73-s − 48·77-s + 1.36e3·79-s − 782·83-s + ⋯ |
| L(s) = 1 | − 0.715·5-s − 1.29·7-s + 0.0548·11-s − 0.682·13-s − 0.798·17-s + 2.22·19-s − 0.834·23-s + 0.759·25-s + 2.15·29-s − 2.17·31-s + 0.927·35-s − 1.54·37-s − 1.18·41-s + 0.283·43-s + 0.707·47-s + 0.125·49-s − 0.393·53-s − 0.0392·55-s + 1.50·59-s + 0.235·61-s + 0.488·65-s + 0.641·67-s − 3.03·71-s − 0.920·73-s − 0.0710·77-s + 1.93·79-s − 1.03·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2985984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2985984 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 + 8 T - 31 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 24 T + 533 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + p^{3} T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 32 T - 102 T^{2} + 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 56 T + 5858 T^{2} + 56 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 184 T + 20994 T^{2} - 184 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 p T + 21698 T^{2} + 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 336 T + 75814 T^{2} - 336 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 376 T + 87501 T^{2} + 376 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 348 T + 126830 T^{2} + 348 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 312 T + 132478 T^{2} + 312 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 80 T + 102402 T^{2} - 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 228 T + 201634 T^{2} - 228 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 152 T + 253337 T^{2} + 152 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 680 T + 355286 T^{2} - 680 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 112 T + 152970 T^{2} - 112 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 352 T + 289170 T^{2} - 352 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1816 T + 1497518 T^{2} + 1816 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 287 T + p^{3} T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 1360 T + 1144350 T^{2} - 1360 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 782 T + 992327 T^{2} + 782 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 240 T + 1328110 T^{2} - 240 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 338 T + 1834899 T^{2} + 338 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.657979658602724198801599336123, −8.655245207804427016561027057342, −7.71865936058435891845395438500, −7.65995581970522236304644594546, −7.04654318111114648694219498127, −6.93824572115817663177308935794, −6.45057268781794376156963981019, −6.00919470476714970975290137666, −5.31980992451034609078984818371, −5.21105601935098106047618011095, −4.64724178061304278728385085557, −4.07034567652829383277988155426, −3.56525411132870476683038887215, −3.29989827448718207981552344211, −2.82851116980328152995142854890, −2.29435112450139826845477479923, −1.52135504784072397550649566365, −0.902024348303270334487324684550, 0, 0,
0.902024348303270334487324684550, 1.52135504784072397550649566365, 2.29435112450139826845477479923, 2.82851116980328152995142854890, 3.29989827448718207981552344211, 3.56525411132870476683038887215, 4.07034567652829383277988155426, 4.64724178061304278728385085557, 5.21105601935098106047618011095, 5.31980992451034609078984818371, 6.00919470476714970975290137666, 6.45057268781794376156963981019, 6.93824572115817663177308935794, 7.04654318111114648694219498127, 7.65995581970522236304644594546, 7.71865936058435891845395438500, 8.655245207804427016561027057342, 8.657979658602724198801599336123
