| L(s) = 1 | − 4·2-s − 3·3-s + 8·4-s − 18·5-s + 12·6-s − 32·8-s + 72·10-s + 50·11-s − 24·12-s − 72·13-s + 54·15-s + 128·16-s − 126·17-s + 72·19-s − 144·20-s − 200·22-s − 14·23-s + 96·24-s + 125·25-s + 288·26-s + 27·27-s + 316·29-s − 216·30-s + 36·31-s − 256·32-s − 150·33-s + 504·34-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s − 1.60·5-s + 0.816·6-s − 1.41·8-s + 2.27·10-s + 1.37·11-s − 0.577·12-s − 1.53·13-s + 0.929·15-s + 2·16-s − 1.79·17-s + 0.869·19-s − 1.60·20-s − 1.93·22-s − 0.126·23-s + 0.816·24-s + 25-s + 2.17·26-s + 0.192·27-s + 2.02·29-s − 1.31·30-s + 0.208·31-s − 1.41·32-s − 0.791·33-s + 2.54·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21609 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21609 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + p^{2} T + p^{3} T^{2} + p^{5} T^{3} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 18 T + 199 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 50 T + 1169 T^{2} - 50 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 36 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 126 T + 10963 T^{2} + 126 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 72 T - 1675 T^{2} - 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 14 T - 11971 T^{2} + 14 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 158 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 36 T - 28495 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 162 T - 24409 T^{2} - 162 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 270 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 324 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 72 T - 98639 T^{2} - 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 22 T - 148393 T^{2} - 22 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 468 T + 13645 T^{2} + 468 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 792 T + 400283 T^{2} + 792 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 232 T - 246939 T^{2} + 232 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 734 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 180 T - 356617 T^{2} + 180 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 236 T - 437343 T^{2} + 236 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 36 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 234 T - 650213 T^{2} + 234 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 468 T + p^{3} T^{2} )^{2} \) |
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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
−12.07653512680595501408641903162, −11.79686476357514307127734961111, −11.53639251519702535670550255020, −10.80167456299345189206957689926, −10.14595375311847695350886023968, −9.794404910288988365472095613191, −9.095293354547065102967212782462, −8.713437068392087788187797026833, −8.280910372127769070634519980474, −7.65761356988255321816198631900, −7.03951605894775838375946492292, −6.61204974425279908054834065742, −6.07962032875324163357809601084, −4.81945259426370886347912121054, −4.54117402065392770454252524608, −3.47096534539461245038430210247, −2.77723989039261325963493674381, −1.33704890239877054658380872613, 0, 0,
1.33704890239877054658380872613, 2.77723989039261325963493674381, 3.47096534539461245038430210247, 4.54117402065392770454252524608, 4.81945259426370886347912121054, 6.07962032875324163357809601084, 6.61204974425279908054834065742, 7.03951605894775838375946492292, 7.65761356988255321816198631900, 8.280910372127769070634519980474, 8.713437068392087788187797026833, 9.095293354547065102967212782462, 9.794404910288988365472095613191, 10.14595375311847695350886023968, 10.80167456299345189206957689926, 11.53639251519702535670550255020, 11.79686476357514307127734961111, 12.07653512680595501408641903162
