| L(s) = 1 | − 6·3-s − 6·5-s + 14·7-s + 27·9-s + 6·11-s − 16·13-s + 36·15-s − 6·17-s − 64·19-s − 84·21-s + 6·23-s − 166·25-s − 108·27-s + 252·29-s + 40·31-s − 36·33-s − 84·35-s + 248·37-s + 96·39-s − 450·41-s − 376·43-s − 162·45-s − 12·47-s + 147·49-s + 36·51-s + 1.10e3·53-s − 36·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.536·5-s + 0.755·7-s + 9-s + 0.164·11-s − 0.341·13-s + 0.619·15-s − 0.0856·17-s − 0.772·19-s − 0.872·21-s + 0.0543·23-s − 1.32·25-s − 0.769·27-s + 1.61·29-s + 0.231·31-s − 0.189·33-s − 0.405·35-s + 1.10·37-s + 0.394·39-s − 1.71·41-s − 1.33·43-s − 0.536·45-s − 0.0372·47-s + 3/7·49-s + 0.0988·51-s + 2.86·53-s − 0.0882·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 6 T + 202 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 6 T + 1246 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 16 T + 2406 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 9778 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 64 T + 6534 T^{2} + 64 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 7870 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 252 T + 56446 T^{2} - 252 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 40 T - 13890 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 248 T + 98214 T^{2} - 248 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 450 T + 175642 T^{2} + 450 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 376 T + 161526 T^{2} + 376 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 12 T + 141790 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 1104 T + 602230 T^{2} - 1104 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 804 T + 380614 T^{2} + 804 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 428 T + 425886 T^{2} - 428 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 148 T + 440790 T^{2} + 148 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 954 T + 13106 p T^{2} - 954 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 1072 T + 1063278 T^{2} - 1072 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 572 T + 901662 T^{2} + 572 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1944 T + 1957030 T^{2} + 1944 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 366 T + 1156090 T^{2} - 366 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 808 T + 903054 T^{2} - 808 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.935875836800212370276937823398, −8.570875187514156305629976137112, −8.048083388563251678082769578685, −8.019008158270384535428095562060, −7.33324929655751739078252772369, −6.91664702399053509525739611420, −6.51525544320108285545347014859, −6.29082122808320892733657556500, −5.47810956510678635547126100971, −5.40113170974896967325517807949, −4.71194948360299698001188611618, −4.54014900282654338976182976370, −3.78306799873468030987411495960, −3.75426907223873680110850399619, −2.57632016609236210813371187859, −2.34294645760787643412791071487, −1.37353813189303629459664038798, −1.12283065357813089825443764282, 0, 0,
1.12283065357813089825443764282, 1.37353813189303629459664038798, 2.34294645760787643412791071487, 2.57632016609236210813371187859, 3.75426907223873680110850399619, 3.78306799873468030987411495960, 4.54014900282654338976182976370, 4.71194948360299698001188611618, 5.40113170974896967325517807949, 5.47810956510678635547126100971, 6.29082122808320892733657556500, 6.51525544320108285545347014859, 6.91664702399053509525739611420, 7.33324929655751739078252772369, 8.019008158270384535428095562060, 8.048083388563251678082769578685, 8.570875187514156305629976137112, 8.935875836800212370276937823398
