| L(s) = 1 | + 2·3-s − 2·5-s − 6·7-s + 3·9-s − 2·13-s − 4·15-s − 2·19-s − 12·21-s − 4·23-s − 2·25-s + 4·27-s − 8·29-s − 14·31-s + 12·35-s − 4·39-s − 14·41-s − 4·43-s − 6·45-s − 8·47-s + 18·49-s − 8·53-s − 4·57-s − 4·59-s + 16·61-s − 18·63-s + 4·65-s + 2·67-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s − 2.26·7-s + 9-s − 0.554·13-s − 1.03·15-s − 0.458·19-s − 2.61·21-s − 0.834·23-s − 2/5·25-s + 0.769·27-s − 1.48·29-s − 2.51·31-s + 2.02·35-s − 0.640·39-s − 2.18·41-s − 0.609·43-s − 0.894·45-s − 1.16·47-s + 18/7·49-s − 1.09·53-s − 0.529·57-s − 0.520·59-s + 2.04·61-s − 2.26·63-s + 0.496·65-s + 0.244·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1557504 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1557504 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 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 | $C_4$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 + 14 T + 106 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 14 T + 126 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 90 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_4$ | \( 1 - 16 T + 166 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2 T + 90 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 122 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 18 T + 214 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 24 T + 318 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
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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
−9.476155761854977191581152069695, −9.158617020304468656454575706792, −8.635111220184842423338907541804, −8.440922751915511202953840138079, −7.78057841653783582279308940097, −7.48809025159988600485210646224, −6.99974236517206921538421942345, −6.88982826978935284131741573283, −6.22629618324356178308125877210, −5.80997790946729817082150608454, −5.31114286938088814559651855698, −4.57416808638963492783617023369, −4.04204117058954654068846096471, −3.61941605358019194725051246803, −3.24235093858576376259035646041, −3.13518996426730008441148532214, −2.03561646954228996106126366448, −1.84530197514726606822112934978, 0, 0,
1.84530197514726606822112934978, 2.03561646954228996106126366448, 3.13518996426730008441148532214, 3.24235093858576376259035646041, 3.61941605358019194725051246803, 4.04204117058954654068846096471, 4.57416808638963492783617023369, 5.31114286938088814559651855698, 5.80997790946729817082150608454, 6.22629618324356178308125877210, 6.88982826978935284131741573283, 6.99974236517206921538421942345, 7.48809025159988600485210646224, 7.78057841653783582279308940097, 8.440922751915511202953840138079, 8.635111220184842423338907541804, 9.158617020304468656454575706792, 9.476155761854977191581152069695
