| L(s) = 1 | − 2-s − 11·4-s + 3·5-s − 9·7-s + 15·8-s − 3·10-s − 80·11-s − 26·13-s + 9·14-s + 61·16-s − 19·17-s − 84·19-s − 33·20-s + 80·22-s − 196·23-s − 239·25-s + 26·26-s + 99·28-s + 44·29-s − 86·31-s − 89·32-s + 19·34-s − 27·35-s + 209·37-s + 84·38-s + 45·40-s + 230·41-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 1.37·4-s + 0.268·5-s − 0.485·7-s + 0.662·8-s − 0.0948·10-s − 2.19·11-s − 0.554·13-s + 0.171·14-s + 0.953·16-s − 0.271·17-s − 1.01·19-s − 0.368·20-s + 0.775·22-s − 1.77·23-s − 1.91·25-s + 0.196·26-s + 0.668·28-s + 0.281·29-s − 0.498·31-s − 0.491·32-s + 0.0958·34-s − 0.130·35-s + 0.928·37-s + 0.358·38-s + 0.177·40-s + 0.876·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 p^{2} T^{2} + p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 3 T + 248 T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 9 T + 192 T^{2} + 9 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 80 T + 3650 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 19 T + 8688 T^{2} + 19 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 84 T + 11130 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 196 T + 33326 T^{2} + 196 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 44 T + 10094 T^{2} - 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 86 T + 56518 T^{2} + 86 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 209 T + 112120 T^{2} - 209 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 230 T + 149010 T^{2} - 230 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 287 T + 92698 T^{2} - 287 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 435 T + 192728 T^{2} + 435 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 118 T + 297410 T^{2} - 118 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 368 T + 379266 T^{2} - 368 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 1058 T + 580378 T^{2} + 1058 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 68 T + 373930 T^{2} - 68 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 131 T + 493328 T^{2} - 131 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 456 T + 542718 T^{2} - 456 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1008 T + 1233294 T^{2} + 1008 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1958 T + 1961238 T^{2} + 1958 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 720 T + 899726 T^{2} - 720 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 928 T + 943870 T^{2} + 928 p^{3} T^{3} + p^{6} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.82561550606998652171882579246, −12.58901836356098803197472894463, −11.83983175074435025619761270369, −11.06425850649763885701510454204, −10.43140371193838474581833723237, −10.09807116794461824291465021047, −9.561810872571921636712154273266, −9.286775519692529610059307449873, −8.292856303885674255776755181762, −8.018015394122309221217538723485, −7.66655583271470656933999512170, −6.61946123359073797781207247279, −5.64369465695127682206552627180, −5.56529669345699534903061496502, −4.36648649967123802853351447847, −4.20795793242617387384424485764, −2.86923266026731074348834312157, −2.05116396334526806829885490341, 0, 0,
2.05116396334526806829885490341, 2.86923266026731074348834312157, 4.20795793242617387384424485764, 4.36648649967123802853351447847, 5.56529669345699534903061496502, 5.64369465695127682206552627180, 6.61946123359073797781207247279, 7.66655583271470656933999512170, 8.018015394122309221217538723485, 8.292856303885674255776755181762, 9.286775519692529610059307449873, 9.561810872571921636712154273266, 10.09807116794461824291465021047, 10.43140371193838474581833723237, 11.06425850649763885701510454204, 11.83983175074435025619761270369, 12.58901836356098803197472894463, 12.82561550606998652171882579246
