| L(s) = 1 | − 2-s − 7·4-s − 22·5-s + 36·7-s + 7·8-s + 22·10-s − 30·11-s − 54·13-s − 36·14-s − 7·16-s + 12·17-s − 38·19-s + 154·20-s + 30·22-s − 244·23-s + 146·25-s + 54·26-s − 252·28-s − 50·29-s + 118·31-s + 71·32-s − 12·34-s − 792·35-s − 218·37-s + 38·38-s − 154·40-s + 50·41-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 7/8·4-s − 1.96·5-s + 1.94·7-s + 0.309·8-s + 0.695·10-s − 0.822·11-s − 1.15·13-s − 0.687·14-s − 0.109·16-s + 0.171·17-s − 0.458·19-s + 1.72·20-s + 0.290·22-s − 2.21·23-s + 1.16·25-s + 0.407·26-s − 1.70·28-s − 0.320·29-s + 0.683·31-s + 0.392·32-s − 0.0605·34-s − 3.82·35-s − 0.968·37-s + 0.162·38-s − 0.608·40-s + 0.190·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29241 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29241 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + T + p^{3} T^{2} + p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 22 T + 338 T^{2} + 22 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 36 T + 878 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 30 T + 1270 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 54 T + 4298 T^{2} + 54 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 12 T + 5110 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 244 T + 34466 T^{2} + 244 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 50 T + 48578 T^{2} + 50 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 118 T + 57486 T^{2} - 118 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 218 T + 27354 T^{2} + 218 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 50 T + 128930 T^{2} - 50 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 196 T + 6918 T^{2} - 196 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 792 T + 353770 T^{2} + 792 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 486 T + 320866 T^{2} - 486 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 588 T + 335494 T^{2} + 588 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 64 T + 453798 T^{2} + 64 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 44 T + 9462 T^{2} + 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 684 T + 671086 T^{2} - 684 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 172 T + 696198 T^{2} + 172 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 918 T + 1172702 T^{2} + 918 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 838 T + 1113182 T^{2} + 838 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 58 T + 597362 T^{2} + 58 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 1340 T + 2139078 T^{2} - 1340 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−11.97749498039939657104695334834, −11.71914025757917436630678740713, −11.02726168946514620089721495846, −10.76423384845950321898562899806, −9.856359351089960075952652037148, −9.715819217355879721618271191367, −8.561726620060260100153365420473, −8.396958718654910878786282949493, −7.944286613760277740991374618219, −7.70676658988963583880047064687, −7.18348145635653684649664533486, −6.14769300649364828788399431732, −5.01943294774810114720715099076, −4.99119447842403901755889422614, −4.09725800651377916074404694064, −3.97738085321924039017203405211, −2.62437641522316407940005515926, −1.63713172301278081567083567993, 0, 0,
1.63713172301278081567083567993, 2.62437641522316407940005515926, 3.97738085321924039017203405211, 4.09725800651377916074404694064, 4.99119447842403901755889422614, 5.01943294774810114720715099076, 6.14769300649364828788399431732, 7.18348145635653684649664533486, 7.70676658988963583880047064687, 7.944286613760277740991374618219, 8.396958718654910878786282949493, 8.561726620060260100153365420473, 9.715819217355879721618271191367, 9.856359351089960075952652037148, 10.76423384845950321898562899806, 11.02726168946514620089721495846, 11.71914025757917436630678740713, 11.97749498039939657104695334834
