| L(s) = 1 | − 2·3-s + 2·5-s + 2·7-s + 2·9-s − 8·11-s − 2·13-s − 4·15-s − 4·17-s + 2·19-s − 4·21-s + 4·23-s − 2·25-s − 6·27-s − 12·29-s + 16·33-s + 4·35-s − 4·37-s + 4·39-s − 4·41-s + 4·45-s − 8·47-s + 3·49-s + 8·51-s − 16·55-s − 4·57-s − 14·59-s + 2·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s + 0.755·7-s + 2/3·9-s − 2.41·11-s − 0.554·13-s − 1.03·15-s − 0.970·17-s + 0.458·19-s − 0.872·21-s + 0.834·23-s − 2/5·25-s − 1.15·27-s − 2.22·29-s + 2.78·33-s + 0.676·35-s − 0.657·37-s + 0.640·39-s − 0.624·41-s + 0.596·45-s − 1.16·47-s + 3/7·49-s + 1.12·51-s − 2.15·55-s − 0.529·57-s − 1.82·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3211264 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3211264 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 34 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_4$ | \( 1 + 12 T + 74 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 14 T + 162 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 2 T + 118 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 12 T + 150 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 94 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 14 T + 210 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 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
−9.350192855167780744727517324046, −8.639317914738830253527066503883, −8.226918823380681324938839163272, −7.77185370798865268721872065279, −7.53112219730028550957823463418, −7.15685392420471898896199678347, −6.63506693508790279560739365532, −6.18997609759406507955268760571, −5.51917579658935556738798755969, −5.48289574234310409620671623617, −5.23061835018876204019277200896, −4.79589178773493400445909396160, −4.29239920115117363035585128496, −3.63971843644313591057156594938, −2.91077978729400868052941694003, −2.47547224727555636269706555927, −1.86105559406579022162662483233, −1.51218650745078128786791197455, 0, 0,
1.51218650745078128786791197455, 1.86105559406579022162662483233, 2.47547224727555636269706555927, 2.91077978729400868052941694003, 3.63971843644313591057156594938, 4.29239920115117363035585128496, 4.79589178773493400445909396160, 5.23061835018876204019277200896, 5.48289574234310409620671623617, 5.51917579658935556738798755969, 6.18997609759406507955268760571, 6.63506693508790279560739365532, 7.15685392420471898896199678347, 7.53112219730028550957823463418, 7.77185370798865268721872065279, 8.226918823380681324938839163272, 8.639317914738830253527066503883, 9.350192855167780744727517324046
