| L(s) = 1 | + 6·3-s − 18·5-s − 10·7-s + 27·9-s + 4·11-s + 26·13-s − 108·15-s − 52·17-s − 142·19-s − 60·21-s − 280·23-s + 10·25-s + 108·27-s − 424·29-s − 178·31-s + 24·33-s + 180·35-s + 136·37-s + 156·39-s − 226·41-s − 72·43-s − 486·45-s + 176·47-s − 186·49-s − 312·51-s − 788·53-s − 72·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.60·5-s − 0.539·7-s + 9-s + 0.109·11-s + 0.554·13-s − 1.85·15-s − 0.741·17-s − 1.71·19-s − 0.623·21-s − 2.53·23-s + 2/25·25-s + 0.769·27-s − 2.71·29-s − 1.03·31-s + 0.126·33-s + 0.869·35-s + 0.604·37-s + 0.640·39-s − 0.860·41-s − 0.255·43-s − 1.60·45-s + 0.546·47-s − 0.542·49-s − 0.856·51-s − 2.04·53-s − 0.176·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97344 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97344 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 18 T + 314 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 10 T + 286 T^{2} + 10 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 T - 666 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 52 T + 8054 T^{2} + 52 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 142 T + 16702 T^{2} + 142 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 280 T + 41486 T^{2} + 280 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 424 T + 93110 T^{2} + 424 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 178 T + 48990 T^{2} + 178 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 136 T + 56358 T^{2} - 136 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 226 T + 144474 T^{2} + 226 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 72 T + 28662 T^{2} + 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 176 T - 29410 T^{2} - 176 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 788 T + 451902 T^{2} + 788 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 728 T + 542166 T^{2} - 728 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 736 T + 564838 T^{2} + 736 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 1054 T + 684622 T^{2} - 1054 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 660 T + 721294 T^{2} + 660 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 24 T - 59650 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 40 T - 308514 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1344 T + 1516550 T^{2} - 1344 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 314 T + 619250 T^{2} - 314 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 264 T + 1841070 T^{2} + 264 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.83394782654989645206140862298, −10.79479718180286426519951884760, −9.930619319756160097953807829626, −9.472805026810281246778095833868, −9.147686717995460193798278020972, −8.486485898348483876344336995153, −7.985894070032192047860297427511, −7.965347053620023151782253387617, −7.32760487574813187890454752491, −6.72377644769637053377010965107, −6.17343318526163125708986232958, −5.63734182546104978125567275658, −4.46759038480210206541089976156, −4.14240725570613377864073640081, −3.60831962071639935710322882913, −3.45709916490424119660510429234, −2.00893613230940240624886554959, −2.00838632588726881683853527300, 0, 0,
2.00838632588726881683853527300, 2.00893613230940240624886554959, 3.45709916490424119660510429234, 3.60831962071639935710322882913, 4.14240725570613377864073640081, 4.46759038480210206541089976156, 5.63734182546104978125567275658, 6.17343318526163125708986232958, 6.72377644769637053377010965107, 7.32760487574813187890454752491, 7.965347053620023151782253387617, 7.985894070032192047860297427511, 8.486485898348483876344336995153, 9.147686717995460193798278020972, 9.472805026810281246778095833868, 9.930619319756160097953807829626, 10.79479718180286426519951884760, 10.83394782654989645206140862298
