| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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
−10.79362011636077056360559561261, −10.55314252260507057026967221669, −9.851108601031503344667657035441, −9.808679348015717556555181655322, −9.035669858805245242311123142654, −8.785405480726387599516454570238, −7.82202061269941747277692550723, −7.59989857386556468797913429421, −6.73022719050793100184404417295, −6.46557243205676936061600349675, −6.04793192761779134497314478525, −5.56055115271339643577045628393, −4.98150113830503365689327347887, −4.40433686169121043609948429682, −3.61100062040112321213645830065, −3.20033352580597576395212986654, −1.82745178827283682517970098341, −1.65353758962625653150982339879, 0, 0,
1.65353758962625653150982339879, 1.82745178827283682517970098341, 3.20033352580597576395212986654, 3.61100062040112321213645830065, 4.40433686169121043609948429682, 4.98150113830503365689327347887, 5.56055115271339643577045628393, 6.04793192761779134497314478525, 6.46557243205676936061600349675, 6.73022719050793100184404417295, 7.59989857386556468797913429421, 7.82202061269941747277692550723, 8.785405480726387599516454570238, 9.035669858805245242311123142654, 9.808679348015717556555181655322, 9.851108601031503344667657035441, 10.55314252260507057026967221669, 10.79362011636077056360559561261
